2. Preliminaries

Graphs. By default, all graphs in this paper are undirected and simple. We occasionally also consider directed graphs, as well as graphs with self-loops. The order of a graph, $v(G)$ , is the number of vertices, and its size, $e(G)$ , is the number of edges. If two vertices $u,v \in V(G)$ are adjacent, we denote this by $u \sim _G v$ and resp. $u \to _G v$ (in the undirected, resp. directed case). The subscript $G$ is omitted when the graph is clear from the context.

Associated with every order- $n$ graph $G = ([n], E)$ is its adjacency matrix:

\begin{equation*} A_G \in M_n({\mathbb{R}}), \text{ where } \forall i,j \in [n]\,:\, (A_G)_{i,j} = \unicode {x1D7D9} \left \{ i \sim j \right \} \ \ \text{(respectively, $i \to j$, if $G$ is directed)} \end{equation*}

The spectrum of a graph, denoted $\textrm {spec}(G)$ , is the multiset of eigenvalues corresponding to its adjacency matrix. If $G$ is undirected, its spectrum is real (since $A_G$ is symmetric). The multiplicity of an eigenvalue $\lambda$ in a (diagonalisable) matrix $A$ is denoted by $\mu _A(\lambda )$ . The rank of a graph is the real rank of its adjacency.

The subgraph induced by a set of vertices $S \subseteq V(G)$ is denoted $G[S]$ . The clique number, the independence number, and chromatic number of a simple graph $G$ are denoted $\omega (G)$ , $\alpha (G)$ and $\chi (G)$ , respectively. We denote the complement of a graph $G$ , by $\overline {G}$ . The adjacency matrix of $\overline {G}$ is $J - (A_G + I)$ . Observe that

\begin{equation*} \big | {\textrm {rank}}(\overline {G}) - {\textrm {rank}}(A_G + I) \big | \leqslant 1, \end{equation*}

therefore, where an additive error of $\pm 1$ is insignificant, we sometimes refer to ${\textrm {rank}}(A_G+I)$ as the ‘complement rank’ of $G$ .

A blowup of a graph $G$ is a graph attained by replacing each vertex by a nonempty anticlique and each edge by a complete bipartite graph. If $w(i)$ is the size of the anticlique that replaces vertex $i$ , we denote the resulting graph by $G^w$ . For matrices, the blowup $A^w$ of a matrix $A$ replaces each entry $A_{i,j}$ with a $(w(i) \times w(j))$ -block of entries $A_{i,j}$ (so $A_{G^w} = A_G^w$ ). Importantly, ${\textrm {rank}}(A^w) = {\textrm {rank}}(A)$ for any $A$ and $w$ .

Two vertices are called twins if they are non-adjacent and have the same set of neighbours. Discarding twins from a graph is called a reduction, and is the inverse operation to blowup. Both blowup and reduction affect neither the chromatic number, nor the rank of a graph.

Boolean functions. It is well known that any Boolean function $f\,:\, \{0,1\}^n \to \{0,1\}$ is uniquely representable as a multilinear polynomial over the Reals, viz.,

\begin{equation*} f(x) = \sum _{S \subseteq [n]} a_S \prod _{i \in S} x_i \in {\mathbb{R}}[x_1, \ldots , x_n] \end{equation*}

As usual, the support of $f$ is the index set of the non-zero coefficients in this representation, denoted $\textrm {mon}(f) {\,:\!=\,} \{ S \,:\, \emptyset \ne S \subseteq [n], a_S \ne 0\}$ . We denote the size of $f$ ’s support by $\textrm {spar}(f) {\,:\!=\,} |\textrm {mon}(f)|$ , and the degree of $f$ is $\deg (f) {\,:\!=\,} \max \{|S| \,:\, S \in \textrm {mon}(p)\}$ . By convention, the constant functions $f \equiv 0$ and $f \equiv 1$ are defined to have degree $0$ .

Matrices. As usual, the identity matrix is denoted $I$ , and the all-ones matrix $J$ . The trace of a square matrix is the sum of entries on its main diagonal. Aside from the regular matrix product, we use the Kronecker and Hadamard products which we now recall. Let $A,C \in M_{m \times n}({\mathbb{R}})$ , and $B \in M_{k \times l}({\mathbb{R}})$ be three matrices. The Kronecker product $A \otimes B \in M_{mk \times nl}({\mathbb{R}})$ and Hadamard (element-wise) product $A \odot C \in M_{m \times n}({\mathbb{R}})$ are defined by:

\begin{equation*} (A \otimes B)_{i,j,x,y} {\,:\!=\,} A_{i,j} B_{x,y} , \text{ and } (A \odot C)_{i,j} {\,:\!=\,} A_{i,j} C_{i,j} \end{equation*}

It is well known that rank is multiplicative under the Kronecker product, and sub-multiplicative under the Hadamard product. We also require the standard Kronecker and the Strong graph products.

Unlike the usual notion of matrix minors, Generalised Minors allow repeated indices.

Definition 2.2. Let $A \in M_{k}({\mathbb{R}})$ be a matrix. The Generalised Minor of $A$ corresponding to the indices $(i_1, \ldots , i_d) \in [k]^d$ is the matrix $A[i_1, \ldots , i_d] \in M_{d}({\mathbb{R}})$ defined by

\begin{equation*} \left (A[i_1, \ldots , i_d]\right )_{s,t} {\,:\!=\,} A_{i_s, i_t}, \end{equation*}

for every $s,t \in [d]$ .

Note that the rank of a generalised minor cannot exceed the rank of its ‘parent’ matrix.

Proposition 2.3. If $A \in M_{k}({\mathbb{R}})$ be a matrix and $(i_1, \ldots , i_d) \in [k]^d$ are indices, then

\begin{equation*}{\textrm {rank}}(A[i_1, \ldots , i_d]) \leqslant {\textrm {rank}}(A)\end{equation*}

Communication complexity. We follow the standard notation and terminology in communication complexity, as introduced in the excellent book [Reference Kushilevitz and Nisan38]. Let $A \in M_{m \times n}({\mathbb{R}})$ be a binary matrix. A combinatorial rectangle of $A$ is a minor with row set $R \subseteq [m]$ and column set $C \subseteq [n]$ . For a bit $b \in \{0,1\}$ , a $b$ -rectangle cover of $A$ is a set of $b$ -monochromatic combinatorial rectangles whose union comprises all $b$ -entries in $A$ . The $b$ -cover number of $A$ , and its $b$ -nondeterministic communication, are defined as follows:Footnote ⁸

\begin{equation*} {\chi}^{b}(A) {\,:\!=\,} {\min} _{\mathcal{R}} |\mathcal{R}|, \text{ and } \mathrm{N}^{\mathrm{b}}(A) {\,:\!=\,} \lceil \log {\chi} ^{b}(A) \rceil \end{equation*}

where the minimum is taken over all $b$ -rectangle covers of $A$ .

As usual, the deterministic communication complexity of ${\textrm {D}}(A)$ , is the least cost of a deterministic communication protocol computing $A$ (where the cost of a protocol is the maximum, over all inputs, of the number of bits transmitted).

2.1 Matrix representation of lifts

Lifting is a powerful technique in the study of communication complexity, first introduced by Raz and McKenzie [Reference Raz and McKenzie51]. An application of this method starts with two Boolean functions, $f\,:\, \{0,1\}^n \to \{0,1\}$ and $g\,:\, \{0,1\}^b \times \{0,1\}^b \to \{0,1\}$ . Typically $b$ is much smaller than $n$ , and $g$ is commonly called a gadget. The corresponding lift is the composition $f \circ g^n\,:\, (\{0,1\}^{b})^n \times (\{0,1\}^{b})^n \to \{0,1\}$ :

\begin{equation*} (f \circ g^n) \big ( (x_1, y_1), \ldots , (x_n, y_n) \big ) {\,:\!=\,} f \big ( g(x_1, y_1), \ldots , g(x_n, y_n) \big ), \end{equation*}

where $x_i, y_j\in \{0,1\}^b$ .

A lifting theorem establishes a relation between the communication complexity of $f \circ g^n$ , the query complexity of $f$ and some property of $g$ . This is done for a particular choice of query and communication models. The logic behind such theorems is this: Generally speaking it is hard to prove lower bounds for a communication model. Lifting theorems accomplish this task by relying on a lower bound in a query model, which is easier to come by. There are numerous examples of such theorems, c.f. [Reference Chattopadhyay, Koucký, Loff and Mukhopadhyay14, Reference Göös, Lovett, Meka, Watson and Zuckerman26, Reference Göös, Pitassi and Watson28, Reference Göös, Pitassi and Watson29, Reference Raz and McKenzie51].

We take a matrix-theoretic view of lifts. Rather than compose two Boolean functions, we compose a Boolean function with a collection of binary matrices,

Definition 2.4. Let $A_1, \ldots , A_n \in M_{m \times k}({\mathbb{R}})$ be binary matrices and let $f\,:\, \{0,1\}^n \to \{0,1\}$ be a Boolean function. The $f$ -lift of $A_1, \ldots , A_n$ , denoted $f \big ( A_1, \ldots , A_n \big )$ , is the $m^n \times k^n$ Boolean matrix,

\begin{equation*} f \big ( A_1, \ldots , A_n \big )_{i, j} {\,:\!=\,} f\big ( (A_1)_{i_1, j_1}, \ldots , (A_n)_{i_n, j_n} \big ), \end{equation*}

for $i = (i_1,\ldots ,i_n) \in [m]^n$ and $j = (j_1,\ldots ,j_n) \in [k]^n$ .

This notion of lifting binary matrices with a Boolean function $f\,:\, \{0,1\}^n \to \{0,1\}$ is intimately related to the multilinear representation of $f$ over the Reals, and to the Kronecker product of matrices.

Proposition 2.5. Let $A_1, \ldots , A_n \in M_{m \times k}({\mathbb{R}})$ be binary matrices and let $f\,:\, \{0,1\}^n \to \{0,1\}$ be a Boolean function whose multilinear expansion is $f(x) = \sum _{S \subseteq [n]} a_S \cdot \prod _{l \in S} x_l \in {\mathbb{R}}[x_1, \ldots , x_n]$ . Then, the $f$ -lift of $A_1, \ldots , A_n$ can be written as:

\begin{equation*} f \big ( A_1, \ldots , A_n \big ) = \sum _{S \subseteq [n]} a_S \cdot \underset {l \in [n]}{\bigotimes } Z^S_l \end{equation*}

where $J_{m \times k}$ is the $(m \times k)$ all-ones matrix and $Z^S_l = \begin{cases} A_l & l \in S \\ J_{m \times k} & l \notin S \end{cases}$

Proof. By the definition of the Kronecker product, $\forall (i,j) \in [m]^n \times [k]^n$ we have:

\begin{align*} \left (\sum _{S \subseteq [n]} a_S \cdot \underset {l \in [n]}{\bigotimes } \begin{cases} A_l & l \in S \\ J_{m \times k} & l \notin S \end{cases} \right )_{i,j} &= \sum _{S \subseteq [n]} a_S \cdot \prod _{l \in [n]} \left (\begin{cases} A_l & l \in S \\ J & l \notin S \end{cases} \right )_{i_l, j_l} \\ &= \sum _{S \subseteq [n]} a_S \cdot \prod _{l \in S} (A_l)_{i_l, j_l} \\ &= f \big ( (A_1)_{i_1, j_1}, \ldots , (A_n)_{i_n, j_n} \big ) = f \big ( A_1, \ldots , A_n \big )_{i,j} \end{align*}

This relationship implies the following powerful bound on the rank of a lifted matrix.

Lemma 2.6. Let $f\,:\, \{0,1\}^n \to \{0,1\}$ be a Boolean function and let $A_1, \ldots , A_n \in M_{m \times k}({\mathbb{R}})$ be binary matrices. Then,

\begin{equation*} {\textrm {rank}} \left ( f(A_1, \ldots , A_n) \right ) \leqslant \sum _{S \in \textrm {mon}(f)} \prod _{i \in S} {\textrm {rank}}(A_i) \end{equation*}

Proof. Let $\sum _{S \in \textrm {mon}(f)} a_S \prod _{i \in S} x_i$ be the multilinear representation of $f$ over the Reals. By Definition2.4,

\begin{align*} f(A_1, \ldots , A_n) = \sum _{S \in \textrm {mon}(f)} a_S \bigotimes _{i \in [n]} \begin{cases} A_i & i \in S \\ J_{m \times k} & i \notin S \end{cases} \end{align*}

The proof now follows by recalling that ${\textrm {rank}}(J) = 1$ , and that rank is subadditive, and multiplicative under the Kronecker product.

2.2 Rank-Ramsey graphs

The Ramsey number $R(s,t)$ is the smallest $n$ such that every graph $G$ of order $n$ contains either an $s$ -clique or an $t$ -anticlique. In other words, either $\omega (G)\geqslant s$ or $\alpha (G)\geqslant t$ . Observe that ${\textrm {rank}}(A_G + I)\geqslant \alpha (G)$ , since the minor corresponding to an anticlique in $G$ is all-zeros in $A_G$ , and an identity submatrix in $A_G + I$ . This suggests a search for graphs where both the clique number $\omega (G)$ and ${\textrm {rank}}(A_G + I)$ are small. We call graphs with such properties Rank-Ramsey. To proceed, we introduce some notation.

Definition 2.7. For every $d \geqslant 1$ , let $\nu _d\,:\, \mathbb{N} \to {\mathbb{N}}$ be the function

\begin{equation*} \nu _d(n) {\,:\!=\,} \min _{G} {\textrm {rank}}( A_G + I ) \end{equation*}

minimising over all order- $n$ graphs $G$ with $\omega (G) \leqslant d$ .

A similar notion for directed graphs is of interest as well,

Definition 2.8. For every $d \geqslant 1$ , let $\eta _d\,:\, \mathbb{N} \to {\mathbb{N}}$ be the function

\begin{equation*} \eta _d(n) {\,:\!=\,} \min _{A \in M_n({\mathbb{R}})} {\textrm {rank}}( A + I ) \end{equation*}

minimising over all $n \times n$ binary matrices $A$ with zeros on the main diagonal, and such that $(A+I)$ has no $J_{d+1}$ principal minor.

There is a simple relation between Ramsey numbers and the function $\nu _d$ :

Proposition 2.9. For any two positive integers $d$ and $n$ , there holds $n \lt R(d+1, \nu _d(n) + 1)$ .

Proof. If $G$ is a graph attaining $\nu _d(n)$ , then by definition $\omega (G) \lt d+1$ . Also, as mentioned, $\alpha (G) \lt {\textrm {rank}}(A_G + I) + 1 = \nu _d(n) + 1$ . Therefore, the order of $G$ is smaller than $R(d+1, \nu _d(n) + 1)$ , as claimed.

Corollary 2.10 [Reference Lee and Shraibman47]. For any two positive integers $n \gt d$ , there holds

\begin{equation*} \nu _d(n) = \Omega \left ( n^{1/d} \cdot \left ( \frac {\log n}{d}\right )^{\tfrac {d-1}{d}} \right ) \end{equation*}

Proof. Apply the classical bounds on Ramsey numbers, due to Ajtai, Komlós and Szemerédi [Reference Ajtai, Komlós and Szemerédi2].

The following lemma aggregates several useful properties of our two quantities.

Lemma 2.11. For every natural number $d$ , it holds that:

1. 1. Both quantities $\nu _{d}(n)$ and $\eta _{d}(n)$ are non-increasing in $d$ .

2. 2. $\eta _d(n) \leqslant \nu _d (n) \leqslant \eta _d(n)^2$ .

3. 3. Both $\nu _d$ and $\eta _d$ are submultiplicative: $\nu _d(k n) \leqslant k \cdot \nu _d(n)$ , and $\eta _d(k n) \leqslant k \cdot \eta _d(n)$ .

and moreover, for every $d_1, d_2$ and $n_1, n_2$ , we have:

1. 4. $\nu _{d_1 d_2}(n_1 n_2) \leqslant \nu _{d_1}(n_1) \nu _{d_2}(n_2)$ , and similarly $\eta _{d_1 d_2}(n_1 n_2) \leqslant \eta _{d_1}(n_1) \eta _{d_2}(n_2)$ .

Proof. We prove each property in turn,

1. 1. Obvious: If $b\gt a$ , then a graph with no $a$ -clique has no $b$ -clique either.

2. 2. The lower bound is obvious. For the upper bound, let $A$ be the matrix attaining $\eta _d(n)$ and consider the Hadamard product $(A+I) \odot (A+I)^T$ .

3. 3. Let $G$ be a graph attaining $\nu _d(n)$ , and let $H$ be the disjoint union of $k$ copies of $G$ . Clearly $\omega (H) = \omega (G)$ and ${\textrm {rank}}(A_H + I) = k \cdot {\textrm {rank}}(A_G + I)$ . The proof for $\eta _d$ is identical.

4. 4. Let $G_1$ and $G_2$ be graphs attaining the minimum for $\nu _{d_1}(n_1)$ and $\nu _{d_2}(n_2)$ , respectively. Let $\Gamma$ be the graph whose adjacency matrix is $A_\Gamma = (A_{G_1} + I) \otimes (A_{G_2} + I) - I$ . By multiplicativity,

   \begin{equation*} {\textrm {rank}}( A_\Gamma + I) = {\textrm {rank}} \left ( (A_{G_1} + I) \otimes (A_{G_2} + I) \right ) = {\textrm {rank}}(A_{G_1} + I) \cdot {\textrm {rank}}(A_{G_2} + I) \end{equation*}
   But for any clique $S$ in $\Gamma$ , we have $|S| \leqslant \omega (G_1) \cdot \omega (G_2)$ , since its projections on either coordinate are cliques in $G_1$ and $G_2$ , respectively. The same proof applies to $\eta$ .

2.2.1 Rank-Ramsey numbers

Maintaining the analogy with Ramsey numbers, we define the Rank-Ramsey numbers,

Definition 2.12. $R^k(s,t)$ is the smallest integer $N$ such that for every graph of order $N$ ,

\begin{equation*} \omega (G) \geqslant s \,or\, {\mathrm {rank}}(A_G + I) \geqslant t \end{equation*}

Clearly, $(s-1)(t-1) \lt R^k(s,t) \leqslant R(s,t)$ . The lower bound follows by taking the disjoint union of $(t-1)$ copies of $K_{s-1}$ . For small numbers $s$ and $t$ , Rank-Ramsey numbers are strictly smaller than Ramsey numbers. In fact they match the trivial bound $R^k(s,t) = (s-1)(t-1) + 1$ for all $2 \leqslant t \leqslant 5$ and any $1\lt s$ .

Theorem 2.13. For $2 \leqslant t \leqslant 5$ and every $s \gt 1$ , there holds $R^k(s, t) = (s-1)(t-1) + 1$ .

This equality fails for substantially larger values of $t$ . For example, we show later on (Corollary4.6) that $R^k(3, 6l+11) \gt 16l$ for every $l \gt 2$ . For $l=6$ this yields $R^k(3, 47) \gt 96 \gt 93 = 2 \cdot 46 + 1$ . Consequently, $R^k(3,n) \gt R^k(n,3)$ , for every sufficiently large $n$ , and in particular, unlike Ramsey numbers, Rank-Ramsey numbers are not symmetric in their parameters. To wit, it is much harder to keep the rank low than to avoid large cliques.

It is not hard to show that a simple connected graph $G$ has rank $2$ iff it is a complete bipartite graph, i.e., a blowup of an edge. Likewise, ${\textrm {rank}}(A_G) = 3$ if and only if $G$ is a blowup of a triangle. Chang, Huang and Ye [Reference Chang, Huang and Yeh12, Reference Chang, Huang and Yeh13], showed that every connected graph of $\textrm {rank}$ $4$ or $5$ , is the blowup of a graph from an explicit finite list of graphs, $\mathcal{G}_4$ and $\mathcal{G}_5$ . This is, in fact, true in general. Indeed, if a graph has a pair of twin vertices and we remove one of the two, the rank remain unchanged (as does connectivity). Such reductions can be applied repeatedly. Also, note that such a reduction is the inverse of a blowup.

Proposition 2.14. For every positive integer $r$ , there is a finite list of graphs $\mathcal{G}_r$ such that every connected graph of rank $r$ is a blowup of a member of $\mathcal{G}_r$ .

Proof. As shown by Kotlov and Lovász [Reference Kotlov and Lovász36], a twin-free graph of rank $r$ has at most $\mathcal{O}(2^{r/2})$ vertices. The claim now follows with

\begin{equation*} \mathcal{G}_r = \left \{\ \Gamma \,:\, {\textrm {rank}}(A_\Gamma ) = r \text{ and } \Gamma \text{ is twin-free }\right \}. \end{equation*}

An adaptation of the results on $\mathcal{G}_4$ and $\mathcal{G}_5$ leads us to the following lemma.

Lemma 2.15. Let $G$ be a simple graph whose complement is connected. Then,

1. 1. ${\textrm {rank}}(A_G + I) = 4$ if and only if $\,\overline {G}$ is a blowup of a graph in .

2. 2. ${\textrm {rank}}(A_G + I) = r$ for $r \in \{2,3\}$ if and only if $\,\overline {G}$ is a blowup of an $r$ -clique.

Proof. Let $G$ be an $n$ -vertex graph with ${\textrm {rank}}(A_G + I) = r$ where $r \in \{2, 3, 4\}$ , and $\overline {G}$ connected. Then,

\begin{equation*} l {\,:\!=\,} {\textrm {rank}}(A_{\overline {G}}) = {\textrm {rank}}(J - (A_G + I) ) \in \left \{r-1, r, r+1\right \} \end{equation*}

So, by [Reference Chang, Huang and Yeh12, Reference Chang, Huang and Yeh13], $\overline {G} = H^w$ is a blowup of a graph $H \in \mathcal{G}_l$ with some weights $w$ . But then,

\begin{align*} r = {\textrm {rank}}(A_G + I_n) &= {\textrm {rank}}(J_n - J_n + A_G + I_n) \\ &= {\textrm {rank}}(J_n - A_{H^w}) \\ &= {\textrm {rank}}(J_n - A_H^w) \\ &= {\textrm {rank}}( (J_{v(H)}-A_H)^w ) = {\textrm {rank}}( J_{v(H)} - A_H ) \end{align*}

therefore $H\in \mathcal{G}_{r-1} \sqcup \mathcal{G}_r \sqcup \mathcal{G}_{r+1}$ and is such that ${\textrm {rank}}(J - A_H) = r$ . This yields the above lists.

We also require the following lemma.

Lemma 2.16. Let $G$ be an $n$ -vertex graph whose complement is a disjoint union of blowups of cliques and isolated vertices. Then, ${\textrm {rank}}(A_G + I) \cdot \omega (G) \geqslant n$ .

Proof. Let $k$ be the number of connected components in $\overline {G}$ , say

\begin{equation*} \overline {G} = \overline {G}_1 \sqcup \ldots \sqcup \overline {G}_k, \text{ where } \overline {G}_i = K_{l_i}^{w_i} \text{ for every $1 \leqslant i \leqslant k$ } \end{equation*}

(where here, with slight abuse of notation, isolated vertices are denoted by $K_1$ ). The clique number is determined by the blowup weights,

\begin{equation*} \omega (G) = \alpha (\overline {G}) = \sum _{i=1}^k \alpha (\overline {G}_i) = \sum _{i=1}^k \left \lVert w_i\right \rVert _\infty , \end{equation*}

and conversely, denoting $l^\star {\,:\!=\,} \max \left \{ l_i \,:\, 1 \leqslant i \leqslant k \right \}$ and $L {\,:\!=\,} l_1 + \ldots + l_k$ , and letting $w$ be the concatenation of the weights for each of the components, we may bound the rank as follows

\begin{align*} {\textrm {rank}}(A_G + I_n) = {\textrm {rank}}(J_n - A_{\overline {G}}) &= {\textrm {rank}}(J_L^w - A_{K_{l_1}^{w_1} \sqcup \ldots \sqcup K_{l_k}^{w_k}}) \\ &= {\textrm {rank}}(J_L - A_{K_{l_1} \sqcup \ldots \sqcup K_{l_k}}) \\ &\geqslant {\textrm {rank}}(J_{l^\star } - A_{K_{l^\star }}) = {\textrm {rank}}(I_{l^\star }) = l^\star \end{align*}

where the last inequality follows by taking a principal minor. The claim now follows, as

\begin{equation*} n = \sum _{i=1}^k \langle w_i, \unicode {x1D7D9} \rangle \leqslant \sum _{i=1}^k l_i \left \lVert w_i\right \rVert _\infty \leqslant l^\star \sum _{i=1}^k \left \lVert w_i\right \rVert _\infty \leqslant {\textrm {rank}}(A_G + I) \cdot \omega (G) \end{equation*}

The proof of Theorem2.13 now follows.

Proof. Let $G$ be an $n$ -vertex graph with $n \gt 1$ and ${\textrm {rank}}(A_G + I) = t-1$ , and write

\begin{equation*} \overline {G} = \overline {G}_1 \sqcup \ldots \sqcup \overline {G}_k, \text{ and } r_i {\,:\!=\,} {\textrm {rank}}(A_{\overline {G}_i}) \text{ for every $1 \leqslant i \leqslant k$ } \end{equation*}

where $k \geqslant 1$ is the number of components in $\overline {G}$ . Clearly, $r_1 + \ldots + r_k = {\textrm {rank}}(A_{\overline {G}}) = {\textrm {rank}}(J - (A_G + I)) \leqslant t$ . The cases $t = \{2,3,4\}$ are thus handled, since if $k=1$ then $\overline {G}$ is a blowup of a clique (recall Lemma2.15), and if $k \gt 1$ each component is an isolated vertex or blowup of a clique, which is covered by Lemma2.16.

It remains to handle $t=5$ , in the cases where not all the non-trivial components of $\overline {G}$ are blowups of cliques. Let $\mathcal{H}$ be the list of graphs in Lemma2.15, excluding $K_4$ . Only one possibility still remains to be dealt with: $\overline {G}$ is a blowup $H^w$ of some graph $H \in \mathcal{H}$ , together with (possibly) some isolated vertices. We can rule out the existence of isolated vertices, since then:

\begin{align*} {\textrm {rank}}(A_G + I) = {\textrm {rank}}(J - A_{\overline {G}}) = {\textrm {rank}} \left ( \begin{array}{c|c} J - A_{H^w} & J \\ \hline J & J \end{array} \right ) = {\textrm {rank}} \left ( \begin{array}{c|c} J - A_{H} & \unicode {x1D7D9}^T \\ \hline \unicode {x1D7D9} & 1 \end{array} \right ) \end{align*}

and for all $H \in \mathcal{H}$ , the latter rank is $5 \gt {\textrm {rank}}(A_G+I) = t-1 = 4$ , a contradiction.

Finally we deal with the case $\overline {G} = H^w$ . Maximal independent sets in $H^w$ correspond to blowups of anticliques in $H$ . Therefore, the following LP bounds the independence number of blowups of $H$ :

\begin{align*} \begin{array}{ll@{}ll} \text{minimize} & a &\\ \text{subject to}& x \geqslant 0& \\ & \langle \unicode {x1D7D9}, x \rangle = 1 & \\ & \langle \unicode {x1D7D9}_S, x \rangle \leqslant a &\ \ \ \forall S \in I(H) \end{array} \end{align*}

where $I(H)$ is the set of all independent sets of $H$ . Let $\alpha ^{\star }_H$ be the optimum of the above programme. The least optima of this LP over the base-graphs in $\mathcal{H}$ is exactly $1/3$ , by direct computation. Therefore,

\begin{equation*} \frac {\omega (G)}{n} = \frac {\alpha (H^w)}{\langle \unicode {x1D7D9}, w \rangle } \geqslant \alpha ^{\star }_H \geqslant \frac {1}{3} = \frac {1}{{\textrm {rank}}(A_G + I) - 1} \end{equation*}

and the proof follows by re-arranging.

2.2.2 Warmup: Simple constructions of Rank-Ramsey graphs

As Proposition2.9 shows, any explicit (constructive) sublinear bound on $\nu _d(n)$ yields an explicit Ramsey graph. Moreover, the bound on the independence number is replaced by the possibly stricter requirement concerning the rank. How difficult is it to accomplish this? The directed quantity $\eta _1$ is easy to bound. Shigeta and Amano [Reference Shigeta and Amano52] obtained an asymptotically tight estimate, $\eta _1(n) = \Theta (n^{1/2 + o_n(1)})$ . As a brief warmup, we give the following weaker claim.

Claim 2.17. For every positive integer $n$ , there holds $n^{1/2} \leqslant \eta _1(n) \leqslant \mathcal{O} \left ( n^{\log _4 3} \right )$ .

Proof. For the lower bound, if $A$ is the adjacency matrix of a simple directed graph with no $2$ -cycle, then $(A+I) \odot (A+I)^T = I$ , and by sub-multiplicativity indeed ${\textrm {rank}}(A+I) \geqslant n^{1/2}$ . For the upper bound, note the following properties of $A_{D_4}$ the adjacency matrix of the directed $4$ -cycle,

• • ${\textrm {rank}}(A_{D_4} + I) = 3$ .

• • $A_{D_4} + I$ has no $J_2$ principal minor.

Consequently $\eta _1(4) \leqslant 3$ , and the proof follows by repeatedly applying property $4$ of Lemma2.11.

The upper bound can be likewise improved, e.g., to $\eta _1(n) \leqslant \Theta (n^{\log _6 4})$ , as an exhaustive search through digraphs of order $6$ yields $\eta _1(6) = 4$ . Turning our attention to the undirected quantity $\nu _d(n)$ , we note that $\nu _1(n)=n$ (the graph must have no edges). Instead, we provide, as part of the warmup, a trivial construction for $\nu _d(n)$ . Better estimates provided in the sequel require new ideas.

Proposition 2.18. For every two positive integers $d$ and $n$ , we have $\nu _d(n) \leqslant \lceil \frac {n}{d} \rceil$ .

Proof. Let $G$ be the disjoint union of $\lfloor \frac {n}{d} \rfloor$ copies of $K_{d}$ and a single copy of $K_{n \bmod d}$ . By construction, $\omega (G) = d$ . Furthermore, $(A_G + I)$ is a block matrix of $\lceil \frac {n}{d} \rceil$ matrices J, thus ${\textrm {rank}}(A_G + I) = \lceil \frac {n}{d} \rceil$ .

3. Rank-Ramsey graphs and the Log-Rank conjecture

In this section we briefly overview the connections between the log-rank problem in communication complexity, and Rank-Ramsey graphs. This discussion is mostly expository and not very technical.

On the one hand, we observe that any Rank-Ramsey construction witnesses a gap in the graph-theoretic formulation of the log-rank conjecture. Conversely we show that, under certain conditions, exhibiting a gap in the log-rank problem implies the construction of Rank-Ramsey graphs.

3.1 The log-rank conjecture

What is the relation between the rank of a binary matrix and its communication complexity? As observed by Mehlhorn and Schmidt [Reference Mehlhorn and Schmidt49], every deterministic communication protocol induces a partition of the communication matrix into monochromatic rectangles. This clearly implies that ${\textrm {D}}(A) \geqslant \log {\textrm {rank}}(A)$ for every binary matrix $A$ , where ${\textrm {D}}(A)$ is the deterministic communication complexity of $A$ . Does a converse hold? This is one of the most notorious open questions in computational complexity. The log-rank conjecture states that

(1) \begin{equation} {\textrm {D}}(A) \leqslant \mathcal{O} \left ( \log ^c {\textrm {rank}}(A) \right ) \end{equation}

for every binary matrix $A$ , where $c \gt 1$ is some absolute constant.

Our current best lower bound on $c$ is $c \geqslant 2$ , due to Göös, Pitassi and Watson [Reference Göös, Pitassi and Watson28]. The upper bounds that we have remain exponentially far. The current record, due to Sudakov and Tomon [Reference Sudakov and Tomon54] (see also [Reference Lovett45]), states that ${\textrm {D}}(A) = \mathcal{O}({\textrm {rank}}^{1/2}(A))$ .

3.2 The graph-theoretic perspective

There are several known equivalent formulations of the log-rank conjecture, eschewing the use of terminology from communication complexity. For example, Nisan and Wigderson [Reference Nisan and Wigderson50] proved that the conjecture is equivalent to the following statement: Define $\text{mono}(A)\,:\!=\, \max |B|/|A|$ , over all monochromatic minors $B$ of $A$ .Footnote ⁹ The conjecture is that for every binary matrix $A$ there holds

\begin{equation*} -\log \left ( \text{mono}(A) \right ) \leqslant \mathcal{O}( \log ^{c} {\textrm {rank}}(A) ), \end{equation*}

where $c \gt 1$ is a universal constant.

Here, we consider the graph-theoretic formulation, due to Lovász and Saks (see [Reference Lovász and Saks46, Reference Nisan and Wigderson50]), which states that there exists an absolute constant $c\gt 0$ such that for every graph $G$ ,Footnote ¹⁰

\begin{equation*} \log \chi (G) \leqslant \mathcal{O} \left ( \log ^{c} {\textrm {rank}}(A_{G}) \right ) \end{equation*}

To motivate the next definition, let us think of the clique number $\omega (G)$ as the largest cardinality of a set of vertices with independence number $1$ . We consider, likewise, the largest order of an induced subgraph of $G$ , of independence number at most $d$ :

Definition 3.1. Let $G$ be a graph, and $d$ be a positive integer. Then,

\begin{equation*} \psi _d(G) {\,:\!=\,} \max _{\substack {U \subseteq V(G) \\ \alpha (G[U]) \leqslant d}} |U| \end{equation*}

An interesting feature of $\psi _d(G)$ is that it captures the chromatic number, up to log factors. This goes back to the classical results on integrality gaps for covering problems [Reference Lovász43]. A similar argument appears in the analysis [Reference Dietzfelbinger, Hromkovič and Schnitger19], of extended fooling sets.Footnote ¹¹

Claim 3.2. For every graph $G$ of order $n$ , it holds that:

\begin{equation*} \max _{1 \leqslant d \leqslant \alpha (G)} \frac {\psi _d(G)}{d} \leqslant \chi (G) \leqslant \ln n \cdot \max _{1 \leqslant d \leqslant \alpha (G)} \frac {\psi _d(G)}{d} + 1 \end{equation*}

Proof. Let $H$ be an induced subgraph of $G$ attaining $\psi _d(G)$ for some $1 \leqslant d \leqslant \alpha (G)$ . Then,

\begin{equation*} \chi (G)\geqslant \chi (H)\geqslant \frac {\psi _d(G)}{\alpha (H)}\geqslant \frac {\psi _d(G)}{d}. \end{equation*}

The middle inequality follows from the fact that $\alpha (F)\cdot \chi (F)\geqslant v(F)$ for every graph $F$ . The following colouring algorithm yields the upper bound.

• • Let $U_1 \,:\!=\, V(G)$ , and repeat until $U_{i+1}=\emptyset$ :

• • Let $U_{i+1}\,:\!=\,U_i\setminus S_i$ , where $S_i$ is a largest independent set in the induced subgraph $G[U_i]$ .

• • Colour the vertices of $S_i$ with colour $i$ .

If this algorithm terminates after $t$ iterations, then it yields a $t$ -colouring of $G$ , whence $t\geqslant \chi (G)$ . Note also that $|U_i|\leqslant \psi _{|S_i|}(G)$ . Denote $B {\,:\!=\,} \max _{1 \leqslant i \leqslant t} |U_i| / |S_i|$ , i.e., $|S_i| \geqslant \frac {1}{B} |U_i|$ , for $i=1, \ldots , t$ . Thus, for all $i$ , we have

\begin{equation*} |U_{i+1}| = |U_{i}| - |S_i| \leqslant |U_{i}| \cdot \left ( 1 - \frac {1}{B} \right ). \end{equation*}

Iterating, and using this for $i=t-1$ , we get

\begin{equation*} 1 \leqslant |U_t|\leqslant n \cdot \left ( 1 - \frac {1}{B} \right )^{t - 1} \lt n \cdot e^{-\frac {t-1}{B}} \end{equation*}

Taking logs, we have,

\begin{equation*} \chi (G) \leqslant t\lt B\cdot \ln n + 1 \leqslant \max _{1 \leqslant i \leqslant t} \left ( \frac {\psi _{|S_i|}(G)}{|S_i|} \right )\cdot \ln n + 1 \end{equation*}

The following two corollaries connect between the growth rates of the functions $\nu _d(n)$ , and the log-rank conjecture. The first corollary shows that Rank-Ramsey graphs witness a gap in the graph-theoretic formulation of the log-rank conjecture. Conversely, the second corollary shows how under certain circumstances, log-rank separations can imply constructions of Rank-Ramsey graphs (see also Subsection 5.3 for an example of the interplay between the log-rank conjecture and Rank-Ramsey graphs).

Corollary 3.3. For any two positive integers $d$ and $n$ , there exists a graph $G$ such that:

\begin{equation*} \chi (G) \geqslant \frac {n}{d}, \text{ and } {\mathrm {rank}}(A_{G}) \leqslant \nu _d(n) + 1 \end{equation*}

Proof. Let $\overline {G}$ be an order- $n$ graph attaining $\nu _d(n)$ . Then, $\chi (G) \alpha (G) \geqslant n$ and $\alpha (G)=\omega (\overline {G}) \leqslant d$ . And,

\begin{equation*} {\textrm {rank}}(A_{G}) = {\textrm {rank}}(J - A_{\overline {G}} - I) \leqslant {\textrm {rank}}(A_{\overline {G}} + I) + 1 = \nu _d(n) + 1 \end{equation*}

Corollary 3.4. For every graph $G$ , there exists some $1 \leqslant d \leqslant \alpha (G)$ such that,

\begin{equation*} \chi (G) \leqslant \ln n \cdot \frac {\psi _d(G)}{d} + 1, \text{ and } {\mathrm {rank}}(A_G) \geqslant \nu _d(\psi _d(G)) - 1 \end{equation*}

Proof. Let $1 \leqslant d \leqslant \alpha (G)$ be a number for which the upper bound in Claim3.2 holds. This gives us the left inequality. For the right inequality, let $H$ be an induced subgraph of $G$ attaining $\psi _d(G)$ . Then,

\begin{equation*} {\textrm {rank}}(A_G) \geqslant {\textrm {rank}}(A_H) = {\textrm {rank}}(J-A_{\overline {H}}-I) \geqslant {\textrm {rank}}(A_{\overline {H}}+I) - 1 \geqslant \nu _d(\psi _d(G)) - 1 \end{equation*}

4. Rank-Ramsey graphs from minors of Kronecker powers

In this section we obtain a polynomial separation between $\nu _d(n)$ and $n$ , for all $d \geqslant 41$ .

As we know, such gaps are implied by the existence of low-rank Boolean matrices with the desired properties. However, whereas most (non-explicit) constructions of Ramsey graphs found in the literature use the probabilistic method, this avenue is closed for us. This is due to the fact that we have no natural distribution over low-rank Boolean matrices. Consequently, we opt for explicit constructions.

The proof of Theorem4.1 is divided into two parts: first, we find a constant size graph $G$ , beating the trivial bound from Proposition2.18 for some small $d$ . Then, we show how to locate a good principal minor within large Kronecker powers of $(A_G + I)$ , such that the clique number remains low.

4.1 Part 1: Finding a base graph

Our first goal is to find some graph $G$ , such that ${\textrm {rank}}(A_G + I)$ is strictly smaller than $\frac {v(G)}{\omega (G)}$ . It is even better if $G$ has a small clique number (say, $\omega (G)=2$ ), as this improves the bounds in the blowup procedure (of part 2). This is equivalent to seeking a graph where the eigenvalue $-1$ has a large multiplicity, since ${\textrm {rank}}(A_G + I) = v(G) - \mu _{A_G}(-1)$ .

An exhaustive search reveals that there is no such graph with fewer than $10$ vertices. Therefore, we seek larger, highly structured graphs. Strongly-regular graphs are natural candidates for such a search. These are regular, highly symmetric graphs with only three distinct eigenvalues (the Perron eigenvalue, equal to the degree, and two other distinct eigenvalues):

Definition 4.2. Let $v \gt k \gt \max \{\lambda , \mu \} \geqslant 0$ be integers. We say that $G$ is an $\textrm {srg}(v,k,\lambda ,\mu )$ (i.e., strongly regular graph with parameters $v$ , $k$ , $\lambda$ and $\mu$ ) if

1. 1. $G$ has order $v$ .

2. 2. It is $k$ -regular.

3. 3. Every two adjacent vertices in $G$ have $\lambda$ common neighbours.

4. 4. Every two non-adjacent vertices in $G$ have $\mu$ common neighbours. Footnote ¹²

We cannot use strongly regular graphs as base graphs for $d=2$ , because apart from $K_2$ and the complete bipartite graphs $K_{n,n}$ (whose spectrum is symmetric, and thus cannot beat the trivial bound of Proposition2.18), only seven triangle-free strongly regular graphs are known: $C_5$ , Petersen, Clebsch, Hoffman-Singleton, Gewirtz, Mesner, and the Higman-Sims graph. Whether this list is exhaustive or not is a famous open question (Cf. [Reference Biggs8, Reference Brouwer and Haemers7]). But even if such a graph exists, another problem arises.

Proposition 4.3. $G = K_2$ is the only triangle-free strongly-regular graph for which $-1 \in \textrm {spec} (G)$ .

Proof. Say $G$ is in $\textrm {srg}(v, k, \lambda , \mu )$ , let $A$ be its adjacency matrix, and let $r \gt s$ be its two distinct non-Perron eigenvalues. Observe that $\lambda = 0$ since $G$ is triangle-free. The following relation among the parameters of strongly-regular graph are well known (e.g., [Reference Brouwer and Haemers7]):

\begin{equation*} (1)\ \ k-\mu = -rs,\ \ (2)\ \ \lambda = \mu + r + s, \text{ and }\ \ (3)\ \ (v-k-1)\mu = k(k-\lambda -1) \end{equation*}

If $r=-1$ then $(1) \implies k-\mu = s \lt r \lt 0$ , which implies that $k \lt \mu$ , a contradiction. Alternatively, if $s=-1$ , then $(2) \implies \mu = 1-r \lt 2$ and either $\mu = 0$ (and $(3) \implies k=1$ , therefore $G=K_2$ ), or $\mu =1$ (and $r=0$ ). In the latter case, it follows (see [Reference Deutsch and Fisher18]) that the graph must be a Moore graph, since substituting back into $(3)$ yields:

\begin{equation*} v = k^2 + 1 = 1 + k \sum _{i=0}^{\textrm {diam}(G)-1} (k-1)^i, \end{equation*}

thus $G$ is a Moore graph (which is clearly connected of diameter two, as it is strongly-regular). It remains to rule out the Moore graphs. Firstly, $G$ can be neither the complete graph, nor an odd cycle over $\gt 5$ vertices, as they are not strongly-regular. Moreover, $G$ cannot be $C_5$ , the Petersen graph, the Hoffman-Singleton graph, or the famed (hypothesised) $57$ -regular Moore graph, as their spectra are all fully determined (c.f. [Reference Biggs8, Reference Godsil and Royle30]), and do not match $r=0,\, s=-1$ .

Proposition4.3 appears to rule out the use of strongly regular graphs as base graphs, if we insist on $d=2$ . However, we have a way out: A triangle-free strongly regular graph can have the eigenvalue $1$ . This suggests that we find a way to negate the spectrum of such graphs, while retaining triangle-freeness. As the following two simple lemmas show, this is indeed possible.

Lemma 4.4. For any two graphs, $G$ and $H$ , $\omega (G \otimes H) \leqslant \min \{\omega (G), \omega (H)\}$ .

Proof. Let $S \subseteq V(G) \times V(H)$ be a clique in $G \otimes H$ . By definition of the Kronecker product, the projections of $S$ onto $G$ is a clique, and likewise for $H$ . As the graphs are simple, every vertex appears in these projections exactly once.

Lemma 4.5. Let $A$ be the adjacency matrix of a graph $G$ and let $l \gt 2$ be an integer. Then:

\begin{equation*} \mu _{A \otimes K_l} (-1) = (l-1) \cdot \mu _A (1) \end{equation*}

Proof. Recall that the spectrum of the Kronecker Product of two matrices is the pairwise product of their respective eigenvalues. Since $\textrm {spec}(A_{K_l}) = \{ l^{(1)}, -1^{(l-1)}$ }, it follows that

\begin{equation*} \mu _{A \otimes K_l} (-1) = (l-1) \cdot \mu _A (1) + \mu _A \left ( \frac {1}{l-1} \right ) \end{equation*}

but $\mu _A (\frac {1}{l-1}) = 0$ , since $A$ is symmetric and binary, and thus all its eigenvalues are algebraic integers.

In combination, Lemmas4.4 and 4.5 yield some base-graphs and upper bounds on $\nu _2$ , and $\nu _3$ .

Corollary 4.6. For any $l \gt 2$ , we have $\nu _2(16l) \leqslant 6l + 10$ , and $\nu _3(27l) \leqslant 7l + 20$ .

Proof. Let $\mathcal{C}$ be the Clebsch graph, and let $\mathcal{S}$ be the Schläfli graph. It is known that:

\begin{equation*} \omega (\mathcal{C}) = 2,\ \ \mu _{A_{\mathcal{C}}}(1) = 10,\ \ v(\mathcal{C}) = 16 \quad \text{ and }\quad \omega (\overline {\mathcal{S}}) = 3,\ \ \mu _{A_{\overline {\mathcal{S}}}}(1) = 20,\ \ v(\overline {\mathcal{S}}) = 27 \end{equation*}

Consider the graph $G = \mathcal{C} \otimes K_l$ . By construction, $v(G) = v(\mathcal{C}) \cdot v(K_l) = 16l$ , and by Lemma4.5, it follows that $\mu _{A_G}(-1) = \mu _{A_{\mathcal{C}}}(1) \cdot (l-1) = 10(l-1)$ , and therefore ${\textrm {rank}}(A_G + I) = v(G) - \mu _{A_G}(-1) = 6l + 10$ . Moreover, $\mathcal{C}$ is triangle-free, and thus by Lemma4.4, so is $G$ . Similarly $\overline {\mathcal{S}}$ yields the claim about $\nu _3$ .

Remark 4.7. In a very recent paper, Bamberg, Bishnoi, Ihringer and Ravi [Reference Bamberg, Bishnoi and Ihringer5] constructed a family of Cayley graphs of $\mathbb{F}_{2^n}^2$ that improves upon the constant of our triangle-free construction, lowering the bound from $\nu _2(n) \leqslant (3/8 + o(1))n$ (see Corollary 4.6 ), to $\nu _2(n) \leqslant (1/4 + o(1)) n$ .

4.2 Part 2: Principal minors of Kronecker powers

Next we use the base-graphs from Part 1 to construct infinite families of graphs with bounded clique numbers, which exhibit a polynomial separation between the order and rank of the complement. Let $G$ be a base-graph, and let $H$ be the graph with adjacency matrix $(A_G + I)^{\otimes n} - I$ . Clearly $\omega (H) = \omega (G)^n$ (take the direct product of $n$ copies of a largest clique of $G$ , plus self loops). However, as shown below, if $\omega (G)$ is sufficiently small, then there is, nevertheless, a large principal minor of $A_H$ with no large cliques.

Theorem 4.8. Let $G=([k], E)$ be a graph, $d \gt 1$ a positive integer, and $r \,:\!=\, {\textrm {rank}}\left (A_G+I\right )$ . Then, for every large enough $n$ there exists a graph $H$ such that:

1. 1. $\omega (H) \leqslant d$ .

2. 2. ${\mathrm {rank}}(A_H + I) \leqslant r^n$

3. 3. $v(H) = \frac {k^n}{d} \left ( \sum _{t=1}^{\omega (G)} \#K_t(G) \cdot t^{d+1} \right )^{-\frac {n}{d+1}} - 1$ , where $\#K_t(G)$ is the number of copies of $K_t$ in $G$ .

Proof. Let $M = (A_G + I)^{\otimes n}$ , and let $X_1, \ldots , X_N \overset {\mathrm{iid}}{\sim } [k]^n$ be indices in $M$ , sampled independently and uniformly at random (the value of $N$ will be fixed later). We find an upper bound on the probability that the principal (generalised) minor $W = M[X_1, \ldots , X_N]$ of $M$ , contains a principal $(d + 1) \times (d + 1)$ all-ones minor. By union bound:

\begin{equation*} \mathbb{P} \left [ \exists S \in \binom {[N]}{d+1}\,:\, W[\{X_i\}_{i \in S}] = J_{d+1} \right ] \leqslant N^{d+1} \cdot \mathbb{P}_{X_1, \ldots , X_{d+1} \sim [k]^n} \left [ W[X_1, \ldots , X_{d+1}] = J_{d+1} \right ]. \end{equation*}

The choice of $X_1, \ldots , X_{d+1}$ determines a $n \times (d+1)$ matrix with entries $x_{i,j}\in [k]$ .

By definition of the Kronecker product the above event occurs exactly when

\begin{equation*} \forall i,j \in [d+1]\,:\, \forall l \in [n]\,:\, (A_G + I)_{x_{l, i}, x_{l,j}} = 1 \end{equation*}

and by independence, the probability of this event equals

\begin{equation*} \mathbb{P}_{z_1, \ldots , z_{d+1} \sim [k]} \left [ \forall i,j \in [d+1]\,:\, (A_G + I)_{z_i, z_j} = 1 \right ] ^n.\end{equation*}

The latter expression counts the ways to place integer weights on $G$ ’s vertices (with self-loops) such that all vertices in the support (i.e., with non-zero weights) form a clique. This is easily bounded by:Footnote ¹³

\begin{equation*} \mathbb{P}_{z_1, \ldots , z_{d+1} \sim [k]} \big [ \forall i,j \in [d+1]\,:\, (A_G + I)_{z_i, z_j} = 1 \big ] \leqslant \frac {1}{k^{d+1}} \cdot \sum _{t=1}^{\omega (G)} \#K_t(G) \cdot t^{d+1}. \end{equation*}

If this expression is bounded away from $1$ , then a good minor $W$ exists. When this is so, a bound on $N$ follows:

\begin{equation*} N^{d+1} \cdot \left (\frac {1}{k^{d+1}} \cdot \sum _{t=1}^{\omega (G)} \#K_t(G) \cdot t^{d+1} \right )^n \lt 1 \implies N \leqslant \left ( \frac {k^{d+1}}{\sum _{t=1}^{\omega (G)} \#K_t(G) \cdot t^{d+1} } \right )^{\frac {n}{d+1}} \end{equation*}

We can, therefore, fix some values for $X_1, \ldots , X_N$ , such that $W=M[X_1, \ldots , X_N]$ contains no all-ones principal minor of size $d+1$ . By Proposition2.3, ${\textrm {rank}}(W) \leqslant {\textrm {rank}}(M) = r^n$ , since $W$ is a generalised minor of $M$ . The proof is concluded by letting $A_H = W-I$ .

We are now ready to prove Theorem4.1.

Figure 1.

Illustration of the Kronecker product $\mathcal{C} \otimes K_3$ , where $\mathcal{C}$ is the Clebsch graph. Similarly to blowup, vertices are replaced by anticliques, and edges by bipartite graphs. The key difference is that here edges are replaced by complete bipartite graphs minus the identity matching (so the graph is twin-free).

Proof. Fix two constants $d \gt 1$ and $l\gt 2$ . As in the proof of Corollary4.6, let $G = \mathcal{C} \otimes K_l$ , where $\mathcal{C}$ is the Clebsch graph (see Figure 1). As shown there,

\begin{equation*} v(G) = v(\mathcal{C}) \cdot v(K_l) = 16l,\ e(G) = e(\mathcal{C}) \cdot e(K_l) = 40 \binom {l}{2},\text{ and } {\textrm {rank}}(A_G + I) = 6l + 10 \end{equation*}

By applying Theorem4.8 to $G$ , we obtain, for every large $N$ a $K_{d+1}$ -free graph $H$ with:

\begin{equation*} {\textrm {rank}}(A_H+I) \leqslant (6l+10)^N, \text{ and } v(H) = \Theta \left ( \frac {(16 l)^{d+1}}{ 16l + 40 \binom {l}{2} \left (2^{d+1} - 2\right ) } \right )^{\frac {N}{d+1}} \end{equation*}

Denoting $n=v(H)$ and re-arranging, we may write:

\begin{equation*} {\textrm {rank}}(A_H + I) \leqslant \mathcal{O} \left ( n^\delta \right ), \text{ where } \delta = { \frac {(d+1) \ln (6l + 10)}{(d+1) \ln (16l) - \ln \left (16l + 40 \binom {l}{2} (2^{d+1} - 2) \right ) } } \end{equation*}

For the first bound, take $d=41$ and $l=26$ .

For the second bound, note that for any fixed $l$ , the expression for $\delta$ decreases with $d$ .

\begin{equation*} \lim _{d\to \infty }\delta = \lim _{d\to \infty } { \frac {(d+1) \ln (6l + 10)}{(d+1) \ln (16l) - \ln \left (16l + 40 \binom {l}{2} (2^{d+1} - 2) \right ) } } = \frac {\ln (6l+10)}{\ln (16l)-\ln 2}. \end{equation*}

The minimum of this expression over the integers occurs at $l=37$ . This yields,

\begin{equation*} \delta = \frac {\ln (232)}{\ln (296)} + o_d(1) \end{equation*}

as claimed.

5. Rank-Ramsey graphs from lifts of Erdős-Rényi graphs

So far, we have constructed Rank-Ramsey graphs with constant $d$ . In this section we proceed to the range $d = \Theta (\log n)$ , and prove the following theorem:

These are graphs with a logarithmic clique number, and approaching ${\textrm {rank}}(A_G + I)=\mathcal{O}\left (n^{2/3}\right )$ . For example, one can instantiate Theorem5.1 to obtain:

\begin{equation*} \nu _{10^6 \log n}(n) \leqslant \mathcal{O} \left ( n^{\frac {2}{3} + \frac {1}{1000}} \right ) \end{equation*}

The key ingredient in our proof of Theorem5.1 is our matrix-theoretic view of lifting (Subsection 2.1).

5.1 The functions $\textrm {NAE}$ and $\textrm {AE}$

The graphs constructed in Theorem5.1 are the lifts of matrices with the following Boolean functions.

Definition 5.2. The Boolean functions $\textrm {NAE}\,:\, \{0,1\}^3 \to \{0,1\}$ and $\textrm {AE}\,:\, \{0,1\}^3 \to \{0,1\}$ are defined by:

\begin{align*} \textrm {NAE}(x_1, x_2, x_3) {\,:\!=\,} \begin{cases} 0 & x_1 = x_2 = x_3 \\ 1 & otherwise \end{cases}, \, and\, \textrm {AE} {\,:\!=\,} 1 - \textrm {NAE} \end{align*}

The seminal paper of Nisan and Wigderson [Reference Nisan and Wigderson50] exhibited a gap for the log-rank conjecture using recursive compositions of $\textrm {NAE}$ and lifts thereof (with the $\textrm {AND}$ -gadget). For more on this, see Section 7.

Since the $\textrm {NAE}$ function depends only on the Hamming weight of the input, its unique expression as a multilinear polynomial is a linear combination of elementary symmetric functions,

\begin{equation*} \textrm{NAE}(x_1,x_2,x_3) = e_2(x_1, x_2, x_3) - e_1(x_1, x_2, x_3) + 1 = x_1 x_2 + x_2 x_3 + x_1 x_3 - x_1 - x_2 - x_3 + 1. \end{equation*}

5.2 Our construction

It is more convenient to formulate the proof in terms of $\textrm {AE}$ , rather than $\textrm {NAE}$ . Our aim is to bound from above two quantities of the graphs that we construct: (i) Rank, and (ii) Maximum Clique Size.

The rank. The polynomial representation of $\textrm {AE}$ (in particular, its low degree), in conjunction with Lemma2.6, imply that the lift of any three matrices $A_1, A_2, A_3 \in M_k$ ,

\begin{equation*}M {\,:\!=\,} \textrm {AE}(A_1, A_2, A_3) \in M_{k^3}({\mathbb{R}})\end{equation*}

has low rank, namely ${\textrm {rank}}(M) \leqslant 3k^2 + 3k = \mathcal{O}(k^2)$ .

The clique number. In our construction $A_1$ , $A_2$ and $A_3$ are adjacency matrices of simple graphs, $G_1$ , $G_2$ and $G_3$ respectively. This guarantees that $M$ is a symmetric binary matrix with ones on the diagonal. As an illustration (and an aside), this class of graphs includes the trivial construction of Proposition2.18: if $G_1 = K_k$ and $G_2=G_3=\overline {K}_k$ , the resulting lifted graph is a disjoint union of cliques.Footnote ¹⁴

Consider a set of vertices $S = \left \{ (x_1, y_1, z_1), \ldots , (x_s, y_s, z_s) \right \}$ in $V(G_1) \times V(G_2) \times V(G_3)$ . By definition of $\textrm {AE}$ , this set forms a clique in the lifted graph if and only if the graphs (with multiplicities, i.e., generalised minors) induced over the constituent graphs are identical. Therefore, informally, cliques of the lifted graph emerge from correlations between $G_1$ , $G_2$ and $G_3$ . To avoid correlations, we will sample our base graphs i.i.d. from $G(k, 1/2)$ , as in the classical lower bound on diagonal Ramsey numbers.

Large projections. We view the vertices of the lifted graph as lattice points in $[k] \times [k] \times [k]$ . To control the clique size of the lifted graph, we seek to avoid subsets $S \subseteq [k] \times [k] \times [k]$ with small axis projections $|\pi _x(S)|$ , $|\pi _y(S)|$ and $|\pi _z(S)|$ . Since our constituent graphs are random, sets with large projections will guarantee ‘sufficient randomness’, and will therefore be less likely to induce cliques in the lifted graph. This intuition is made concrete in the proof of Theorem5.1. Beforehand, we use a probabilistic argument to show that there exist large subsets of the cube, in which every subset above a certain cardinality has some large axis projection.Footnote ¹⁵

Lemma 5.3. Let $a \gt 0$ and $1 \gt \eta \gt 0$ be constants. For any sufficiently large $k$ , there exists a subset $T \subset [k]^3$ , of cardinality $|T| = \widetilde {\Theta } \left ( k^{3(1-\eta )} \right )$ , such that for every subset $S \subset T$ of cardinality $|S| = a \log k$ ,

\begin{equation*} \max \left \{ |\pi _x(S)|, |\pi _y(S)|, |\pi _z(S)| \right \} \geqslant \eta |S|= \eta a \log k \end{equation*}

Proof. Let $T$ be a sample of $N$ points, chosen i.i.d and uniformly from $[k]^3$ , with $N$ to be determined later. What is the probability that $T$ fails to satisfy the statement in the lemma? Say that a subset $S\subset T$ of cardinality $|S|=a \log k$ is $x$ -bad if $|\pi _x(S)| \leqslant \eta a \log k$ . Similarly define $y$ and $z$ -bad subsets. Call $S$ bad if it is $x, y$ and $z$ -bad. A sample $T$ with no bad subsets is called appropriate. If the expected number of bad subsets $S\subset T$ is negligible, then by Markov’s inequality, whp $T$ is appropriate.

The event that $S$ is bad at a certain axis (e.g., $x$ -bad) is equivalently formulated as follows: We throw $a \log k$ balls independently into $k$ bins, and all our balls fall into at most $\eta a \log k$ bins. The probability of this bad event is

\begin{equation*}\binom {k}{\eta a \log k}\left (\frac {\eta a\log k}{k}\right )^{a\log k}\leqslant k^{(\eta - 1) a \log k + \mathcal{O}(\log \log k)} .\end{equation*}

So, the probability that $S$ is bad is at most

\begin{equation*} k^{3(\eta - 1) a \log k + \mathcal{O}(\log \log k)} .\end{equation*}

Consequently, the expected number of bad subsets that are contained in our random set $T$ is at most

\begin{equation*} \binom {N}{a \log k} k^{3(\eta - 1) a \log k + \mathcal{O}(\log \log k)} .\end{equation*}

This expression is $o(1)$ provided that

\begin{equation*}N=k^{3\left (1-\eta \right )-o(1)}\end{equation*}

in which case we can conclude that there exists an appropriate sample $T$ of size $N$ .

Since $T$ is chosen with repetitions, we have to account for the possibility that the same point is selected more than once. But no point can be chosen more than $a(1-\eta ) \log k$ times. For otherwise any set of $a \log k$ triples containing these duplicates would be bad and render $T$ inappropriate. It follows that there exists an appropriate set $T' \subset [k]^3$ of cardinality at least $\frac {N}{a(1-\eta ) \log k}$ .

Appropriate sets allow us to identify a large principal minor of the lifted graph, in which every set of cardinality $a \log k$ is ‘sufficiently random’. With it, we are ready to present our construction.

Proof of Theorem 5.1. As mentioned, our construction proceeds by sampling three graphs $G_1$ , $G_2$ , and $G_3$ , independently and uniformly at random from $G \left (k, 1/2 \right )$ . Let $A_1$ , $A_2$ and $A_3$ be their respective adjacency matrices, and let $M {\,:\!=\,} \textrm {AE}( A_1, A_2, A_3 )$ be their lift.

Pick $a \gt 0$ and $1\gt \eta \gt 0$ , construct a subset $T \subset [k]^3$ as in Lemma5.3, and consider $W {\,:\!=\,} M[T]$ . Since $G_1$ , $G_2$ and $G_3$ have no self-loops, the main diagonal of $M$ (and hence $W$ as well) is all-ones. Also,

\begin{equation*}{\textrm {rank}}(W) \leqslant {\textrm {rank}}(M) \leqslant 3k^2 + 3k\end{equation*}

in view of Lemma2.6 and the fact that $\textrm {AE}$ has total degree $2$ .

So, we now have the rank under control and move to deal with the clique number. It remains to show that, with positive probability (over the choice of $G_1$ , $G_2$ and $G_3$ ), the matrix $W$ has no all-ones principal minors of order $a \log k$ . Let $X$ be the random variable that counts the number of such principal minors. If its expectation is $\mathbb{E}(X)=o(1)$ , then whp the resulting graph has no $(a \log k)$ -clique.

Consider a subset $S \subset T$ of cardinality $|S|=a\log k$ . What is the probability that $W[S]$ is an all-ones matrix? According to Lemma5.3, $S$ has a large projection on some axis, say $|\pi _x(S)| \geqslant \eta a \log k$ and for convenience we assume that the first $\eta a \log k$ $x$ ’s are all distinct. By definition of the $\textrm {AE}$ function, the adjacency relations among the $\eta a \log k$ distinct vertices of $\pi _x(S)$ and the corresponding vertices of $\pi _y(S),\pi _z(S)$ , must coincide with each other. That is,

\begin{equation*} A_1\left [ x_1, \ldots , x_{\eta a \log k} \right ] = A_2\left [ y_1, \ldots , y_{\eta a \log k} \right ] = A_3\left [ z_1, \ldots , z_{\eta a \log k} \right ]. \end{equation*}

Since $A_1[x_1, \ldots , x_{\eta a \log k}]$ is a proper minor of $A_1$ , with no repeated indices, it is the adjacency matrix of a $G(\eta a \log k, 1/2)$ graph. Therefore, for any fixed choice of $A_2\left [ y_1, \ldots , y_{\eta a \log k} \right ]$ , it holds that the event $A_1\left [ x_1, \ldots , x_{\eta a \log k} \right ] = A_2\left [ y_1, \ldots , y_{\eta a \log k} \right ]$ occurs with probability exactly

\begin{equation*} 2^{-\binom {\eta a \log k}{2}}=k^{-\left (\frac {\eta ^2 a^2}{2} - o(1)\right )\log k} \end{equation*}

Consequently

\begin{equation*} \mathbb{E}(X) \leqslant \binom {|T|}{a\log k} k^{-\left (\frac {\eta ^2 a^2}{2} - o(1)\right )\log k} = k^{ \left ( 3a(1-\eta ) - \frac {\eta ^2 a^2}{2} + o(1) \right ) \log k }. \end{equation*}

It follows that $\mathbb{E}(X) = o(1)$ when $a\gt \frac {6(1-\eta )}{\eta ^2}$ . The proof now follows by fixing

\begin{equation*} n {\,:\!=\,} |T| = \widetilde {\Theta } (k^{3(1 - \eta )} ),\ \ c {\,:\!=\,} \frac {a}{3(1 - \eta )},\ \ \varepsilon {\,:\!=\,} \frac {2\eta }{3(1 - \eta )} \end{equation*}

5.3 Rank-Ramsey graphs with logarithmic clique number

In Subsection 5.2, we construct Rank-Ramsey graphs with logarithmic clique number and polynomial rank, approaching $n^{2/3}$ . Conversely, in the study of ‘classical’ Ramsey graphs, Erdős [Reference Erdös21] showed that asymptotically almost all $n$ -vertex graphs have logarithmic clique and independence numbers, and this is indeed best-possible up to constants (c.f. [Reference Campos, Griffiths, Morris and Sahasrabudhe11, Reference Erdös and Szekeres22, Reference Thomason55]). This contrast between Ramsey and Rank-Ramsey graphs raises the question: what is the growth rate of $\nu _{a \log n}(n)?$

Our construction witnesses that for every constant $a\gt 0$ , there holds $\nu _{a \log n}(n) = \widetilde {\mathcal{O}}(n^{2/3} + K(a))$ , where $K(a)$ is a constant depending on $a$ . Conversely, we remark that if the log-rank conjecture holds, then by Corollary3.3, there exists a universal constant $c \gt 1$ , such that for every constant $a\gt 0$ ,

\begin{equation*} \nu _{a \log n}(n) = \Omega \left ( 2^{\sqrt [c]{\log (\frac {n}{a \log n})}} \right ) \end{equation*}

6. In search of better bounds on $\nu _2(n)$

The Ramsey numbers $R(3,n)$ have fascinated combinatorialists for decades. Culminating a long line of excellent research, Kim [Reference Kim34] proved that $R(3,n)=\Theta (n^2 / \log n)$ . Even the implicit constant is known, up to a factor of $(4 + o(1))$ [Reference Pontiveros, Griffiths and Morris23]. What about $R^k(3,n)$ , or equivalently, $\nu _2(n)$ ? Trivially,

\begin{equation*} \Omega \left ( \sqrt {n \log n} \right ) \leqslant \nu _2(n) \leqslant \left \lfloor \frac {n}{2}\right \rfloor \end{equation*}

where the lower-bound follows from the Ramsey numbers $R(3,n)$ .

The base-graphs used in Section 4, derived from the Clebsch graph, yields an improvement of the upper bound to $\nu _2(n) \leqslant (3/8 + o(1))n$ . If we wish to improve the lower bound to $\nu _2(n) \geqslant \Omega (n^{1/2 + \varepsilon })$ for some positive constant $\varepsilon$ , then of course, bounds on the independence number will not do, as there exist triangle-free graphs with $\alpha (G) = \Omega (\sqrt {n \log n})$ [Reference Ajtai, Komlós and Szemerédi2]. Instead, we turn to consider other graph parameters. More concretely we investigate parameters that are related to orthonormal representations of graphs, on which a considerable body of research exists. This choice is motivated by the fact that the Gram matrix that corresponds to an orthonormal representation of $G$ is reminiscent of $A_G+I$ .

Definition 6.1. An orthonormal representation $\left \{ w_v \in {\mathbb{R}}^N \right \}_{v \in V(G)}$ of a graph $G$ is an assignment of unit vectors to the vertices of $G$ , such that

\begin{align*} \forall v,u \in V(G)\,:\, \langle w_v, w_u \rangle = \begin{cases} 1 & v=u \\ 0 & v \not \sim u \\ \star & {otherwise} \end{cases} \end{align*}

The Gram matrix of the vectors $w_v$ in any orthnormal representation of an $n$ -vertex graph $G$ , is an $(n \times n)$ positive semidefinite matrix $W$ that agrees with $A_G+I$ on the main diagonal, and on $G$ ’s non-edges. It is very suggestive that such matrices can tell us a lot about $A_G + I$ . Two of the most studied measures related to $M$ are the Lovász number $\vartheta (G)$ , and the minimum semidefinite rank, $\textrm {msr}(G)$ , whose definitions we recall below. Like ${\textrm {rank}}(A_G + I)$ , these two quantities are related to constructions of Ramsey graphs. By the well-known ‘sandwich theorem’ (see [Reference Knuth39]), for every graph $G$ there holds

\begin{equation*} \alpha (G) \leqslant \vartheta (G) \leqslant \textrm {msr}(G) \leqslant \chi (\overline {G}) \end{equation*}

How does ${\textrm {rank}}(A_G+I)$ fit into this web of relations? We show the following.

That is, we show separations between ${\textrm {rank}}(A_G + I)$ , and $\vartheta (G)$ and $\textrm {msr}(G)$ .Footnote ¹⁶ Understanding the relations between $\vartheta (G)$ and ${\textrm {rank}}(A_G + I)$ is particularly interesting in view of striking similarities between the two parameters. See Subsection 6.1 for more.

It is conceivable to us that $\vartheta (G) = \mathcal{O}\left ({\textrm {rank}}(A_G + I)\right )$ . For instance, $\vartheta (G) \leqslant {\textrm {rank}}(A_G+I)$ for every graph of order less than $10$ , and we do not know any graph that fails this inequality. We remark that if this indeed holds true, then the results of [Reference Kashin and Konyagin35, Reference Alon and Kahale1] regarding the Lovász number of graphs with bounded independence number, imply a polynomial improvement on the lower bounds on $\nu _d$ , namely we have $\nu _d(n) = \Omega ( n^{2/(d+1)} )$ , versus the weaker bound $\Omega (n^{1/d})$ (see Corollary2.10) that follow purely from Ramsey numbers. We refer the reader to Appendix B for a discussion on the connection between this open problem, and other well-known problems regarding the relationship between the well-known Hoffman’s bound and Cvetković’s bound.

6.1 Lovász number versus Rank

Let us recall the definition of the Lovász number $\vartheta (G)$ of a simple graph $G$ :

Definition 6.3 [Reference Lovász44]. The Lovász number of a graph $G=(V,E)$ is

\begin{equation*} \vartheta (G) {\,:\!=\,} \min _{c, W} \max _{v \in V} \frac {1}{\langle c, w_v \rangle ^2} \end{equation*}

where $W\,:\, V\to {\mathbb{R}}^N$ is an orthonormal representation of $G$ and $c \in {\mathbb{R}}^N$ is a unit vector.

The graph parameters $\vartheta (G)$ and ${\textrm {rank}}(A_G + I)$ share many properties. E.g., both are multiplicative in the strong product (see Definition2.1), and both are upper bounds on the Shannon capacity of a graph.

Proposition 6.4. Let $G$ and $H$ be two graphs. Then,

\begin{equation*} \vartheta (G \boxtimes H) = \vartheta (G) \vartheta (H), \, and\, {\textrm {rank}}(A_{G \boxtimes H} + I) = {\textrm {rank}}(A_G + I)\, {\textrm {rank}}(A_H + I) \end{equation*}

Proof. Lovász proved ([Reference Lovász44], Lemma 2 and Theorem 7) that $\vartheta$ is multiplicative in the strong product. As for the rank, by Definition2.1 and the multiplicativity of rank under the Kronecker product,

\begin{align*} {\textrm {rank}}(A_{G \boxtimes H} + I) &= {\textrm {rank}}( A_G \otimes A_H + A_G \otimes I + I \otimes A_H + I \otimes I ) \\ &= {\textrm {rank}}( (A_G + I) \otimes (A_H + I) ) = {\textrm {rank}}(A_G + I)\, {\textrm {rank}}(A_H + I) \end{align*}

Corollary 6.5 (See also [Reference Lovász44, Reference Haemers31]). For every graph $G$ it holds that

\begin{equation*} \vartheta (G) \geqslant \Theta (G) \text{ and } {\mathrm {rank}}(A_G + I) \geqslant \Theta (G) \end{equation*}

where $\Theta (G) {\,:\!=\,} \sup _k \sqrt [k]{ \alpha (G^{\boxtimes k}) }$ is the Shannon capacity of $G$ .

6.1.1 Best-possible separation of ${\textrm {rank}}(A_G + I)$ from $\vartheta (G)$

The current best known explicit construction of triangle-free Ramsey graphs is due to Alon [Reference Alon3]. An order- $n$ graph $G$ in this family satisfies $\vartheta (G) = \Theta (n^{2/3})$ . As we will presently show, there also holds that ${\textrm {rank}}(A_{G} + I) = n$ . In other words, these are best-possible $\vartheta$ -Ramsey graphs, yet they are worst-possible Rank-Ramsey graphs.

Let us briefly recount Alon’s construction: Let $k \gt 1$ be such that $3 \nmid k$ and letFootnote ¹⁷

\begin{align*} W_0 {\,:\!=\,} \{ x \in \textrm {GF}(2^k) \,:\, \text{ leftmost bit of $x^7$ is $0$ } \},\quad W_1 {\,:\!=\,} \{ x \in \textrm {GF}(2^k) \,:\, \text{ leftmost bit of $x^7$ is $1$ } \}\end{align*}

and

\begin{align*} U_0 {\,:\!=\,} \{ (w, w^3, w^5) \,:\, w \in W_0 \} \subset \mathbb{Z}_2^{3k}, \quad U_1 {\,:\!=\,} \{ (w, w^3, w^5) \,:\, w \in W_1 \} \subset \mathbb{Z}_2^{3k} \end{align*}

Then, $G_k {\,:\!=\,} \textrm {Cay}\, ( {\mathbb{Z}}_2^{3k}, \{ u_0 + u_1 \,:\, u_0 \in U_0, u_1 \in U_1 \} )$ is a graph on $n_k {\,:\!=\,} |{\mathbb{Z}}_2^{3k}|=2^{3k}$ vertices.

Theorem 6.6 [Reference Alon3]. For every $k \gt 1$ with $3 \nmid k$ , $G_k$ is triangle-free, and $\vartheta (G_k) = \Theta ( n_k^{2/3} )$ .

This bound on $\vartheta$ is tight, as shown by Kashin and Konyagin.

Theorem 6.7 [Reference Kashin and Konyagin35]. For every $n$ vertex triangle-free graph $G$ , there holds $\vartheta (G) \geqslant (2n)^{2/3}$ .

Proof. Kashin and Konyagin [Reference Kashin and Konyagin35] (see also [Reference Alon3]) proved $\vartheta (\overline {G}) \leqslant 2^{2/3} n^{1/3}$ for every triangle-free graph. The claim follows, since $\vartheta (G) \vartheta (\overline {G}) \geqslant n$ for every graph of order $n$ (see [Reference Lovász44], Corollary 2).

In the following claim, we show that ${\textrm {rank}}(A_{G_k} + I) = n_k$ , for every admissible $k$ .

Claim 6.8. For every $k \gt 1$ with $3 \nmid k$ , there holds ${\textrm {rank}}(A_{G_k} + I) = n_k$ .

Proof. Since $G_k$ is a Cayley graph of an Abelian group, ${\mathbb{Z}}_2^{3k}$ , its eigenvalues correspond to sums of group characters, evaluated over the generating set. The characters of ${\mathbb{Z}}_2^t$ are the Fourier-Walsh functions, i.e., the parity functions $\chi _D\,:\, Z_2^t \to \{ \pm 1 \}$ , for every set $D \subseteq [t]$ . So, every $D \subseteq [n]$ , yields an eigenvalue $\lambda _D$ of $A_{G_k}$ , where

\begin{equation*} \lambda _D = \sum _{\substack {u_0 \in U_0 \\ u_1 \in U_1}} \chi _D(u_0 + u_1) = \left ( \sum _{u \in U_0} \chi _D(u) \right ) \left ( \sum _{u \in U_1} \chi _D(u) \right ) \end{equation*}

which is an even integer, because $|U_1| = |W_1| = 2^{k-1}$ (see [Reference Alon3]) are even. Consequently, all eigenvalues of $A_{G_k}$ are even, and ${\textrm {rank}}(A_{G_k} + I) = n_k - \mu _{A_{G_k}}(-1) = n_k$ .

6.2 Minimum semidefinite rank

Another obvious point of comparison is the rank. The minimum semidefinite rank, $\textrm {msr}(G)$ , is the smallest rank of an $n \times n$ orthonormal representation matrix of an order- $n$ graph $G$ . It is easy to see that triangle-free graphs have a large $\textrm {msr}(G)$ .Footnote ¹⁸ Deaett [Reference Deaett17] showed that $\textrm {msr}(G) \geqslant n/2$ , for any connected triangle-free graph. In contrast, in Corollary4.6 we prove that $\nu _2(16l) \leqslant 6l + 10$ , which is attained by a family of connected graphs. Therefore,

Corollary 6.9. For any sufficiently large $n$ , there exists an order $n$ triangle-free graph $G$ , with

\begin{equation*} \textrm {msr}(G) \geqslant n/2 \text{ and } {\mathrm {rank}}(A_G + I) = (3/8 + o(1))n \end{equation*}

We do not know whether a converse of the form ${\textrm {rank}}(A_G+I) = \Omega (\textrm {msr}(G))$ holds.

7. The Nisan-Wigderson construction

In a seminal paper, Nisan and Wigderson [Reference Nisan and Wigderson50] constructed an infinite family of symmetric binary matrices $A_k$ of dimension $2^{3^k}$ . These matrices exhibit a separation between log-rank and communication complexity, and show that $c \geqslant \log _2 3$ in the conjectured inequality (1).

As discussed in Section 3, such separations can be related to Rank-Ramsey graphs. Here, we analyse the Nisan-Wigderson matrices from this perspective, and show that these matrices $A_k$ have very large monochromatic principal minors and yield, therefore, poor Rank-Ramsey graphs.

Claim 7.1. For every $k \geqslant 1$ there exist subsets $S,T \subset [2^{3^k}]$ of cardinality $2^{3^k (1-o_k(1))}$ each, such that

\begin{equation*} A_k[S] = \mathbf{0}, \,and\, A_k[T] = J \end{equation*}

We stress that this does not a contradict the discussion of Section 3: Exhibiting a log-rank separation does not preclude a matrix from having a large monochromatic rectangle.

7.1 The construction

The construction of [Reference Nisan and Wigderson50] involves a lift using the $\textrm {AND}$ gadget, with a recursive composition of the function $\textrm {NAE}$ (as in Definition5.2) with itself. To describe their result, we require some notation.

Definition 7.2. For every $k \gt 1$ , the Boolean function $\textrm {NAE}^k\,:\, \{0,1\}^{3^k} \to \{0,1\}$ is defined:

\begin{align*} \textrm {NAE}^k(x) &{\,:\!=\,} \textrm {NAE} \big ( \textrm {NAE}^{k-1}(x_1, \ldots , x_{3^{k-1}}), \textrm {NAE}^{k-1}(x_{3^{k-1} + 1}, \ldots , x_{2 \cdot 3^{k-1}}),\big.\\ &\big.\quad \textrm {NAE}^{k-1}(x_{2 \cdot 3^{k-1} + 1}, \ldots , x_{3^k}) \big ) \end{align*}

and $\textrm {NAE}^1 {\,:\!=\,} \textrm {NAE}$ .

We denote by $A_k$ the symmetric binary matrix $\textrm {NAE}^k \circ \textrm {AND}^{3^k}$ . Moreover:

\begin{equation*} \forall b \in \{0,1\}\,:\, G^{(b)}_k {\,:\!=\,} G \, ( \{ x \in \{0,1\}^{3^k} \,:\, \textrm {NAE}^k(x) = b \}, \{ \{x,y\} \,:\, \textrm {NAE}^k(x \land y) = 1 \} ) \end{equation*}

in other words, $G^{(1)}_k$ (resp. $G^{(0)}_k$ ) is the graph whose adjacency matrix is the principal minors of $A_k$ , induced by those indices for which the main diagonal is $1$ (resp. $0$ ). With this notation, the result of Nisan and Wigderson says:Footnote ¹⁹

Theorem 7.3 [Reference Nisan and Wigderson50]. Let $A_k {\,:\!=\,} \textrm {NAE}^k \circ \textrm {AND}^{3^k} \in M_{3^k}({\mathbb{R}})$ . Then,

\begin{equation*} \log {\mathrm {rank}}(A_k) \leqslant \mathcal{O} (2^k), \text{ and } {\textrm {D}}(A_k) = 3^k (1 - o_k(1)) \end{equation*}

It easy to bound $\text{rank}(A_k)$ : It is well known (e.g., [Reference Knop, Lovett, McGuire and Yuan37]) that ${\textrm {rank}}(f \circ \textrm {AND}^n) = \textrm {spar}(f)$ for any function $f\,:\, \{0,1\}^n \to \{0,1\}$ . So here the rank is the number of monomials appearing in the expansion of $\textrm {NAE}^{k}$ , which can be bounded by a simple inductive argument.

7.2 Finding large monochromatic principal minors

In search of large monochromatic principal minors of $A_k$ , let us first estimate the orders of the two subgraphs, $G^{(0)}_k$ and $G^{(1)}_k$ .

Proposition 7.4. For every $k \geqslant 1$ , we have that:

\begin{equation*} v\left (G^{(0)}_k\right ) = 2^{3^k} \cdot \left (\frac {1}{3} \pm o_k(1) \right ), \, and \, v\left (G^{(1)}_k\right ) = 2^{3^k} \cdot \left (\frac {2}{3} \pm o_k(1) \right ) \end{equation*}

Proof. Let $p_k$ be the probability that a uniformly random $x \sim \{0,1\}^{3^k}$ is in $V(G^{(1)}_k)$ . Then,

\begin{align*} p_k &= \mathbb{P}_{x \sim \{0,1\}^{3^k}} \left [ \left (\textrm {NAE}^k \circ \textrm {AND}^{3^k}\right ) (x) = 1 \right ] \\ &= \mathbb{P}_{x_1, x_2, x_3 \sim \{0,1\}^{3^{k-1}}} \left [ \textrm {NAE} \left (\textrm {NAE}^{k-1}(x_1), \textrm {NAE}^{k-1}(x_2), \textrm {NAE}^{k-1}(x_3) \right ) = 1 \right ] \\ &= 3 p_{k-1}^2(1-p_{k-1}) + 3 p_{k-1} (1-p_{k-1})^2 = 3 p_{k-1} (1-p_{k-1}). \end{align*}

By direct observation we have $p_1 = 3/4$ . Furthermore, $x\gt 3x(1-x)\gt \tfrac 23$ for $\tfrac 34 \gt x \gt \tfrac 23$ . Thus, the sequence $p_k$ is decreasing to its limit, the unique positive root of $x=3x(1-x)$ , namely $x=2/3$ .

We now construct large monochromatic principal minors in $G^{(0)}_k$ and $G^{(1)}_k$ .

Proposition 7.5. The matrices $A_k$ have large monochromatic principal minors. That is,

\begin{equation*} \min \left \{ \omega \left (G^{(1)}_k\right ), \alpha \left (G^{(0)}_k \right ) \right \} \geqslant 2^{3^k \left ( 1 - o_k(1) \right )} \end{equation*}

Proof. For every positive $k$ , let us denote

\begin{equation*} \alpha _k {\,:\!=\,} \alpha \left (G^{(0)}_{k} \right ), \text{ and }\omega _k {\,:\!=\,} \omega \left ( G^{(1)}_k \right ) \end{equation*}

We claim that $\alpha _k \geqslant \omega _{k-1}^3$ and that $\omega _k \geqslant 2^{3^{k-1}} \alpha _{k-1} \omega _{k-1}$ , for any $k \geqslant 2$ . Let $A,B \subset \{0,1\}^{3^{k-1}}$ be a largest clique of $G^{(1)}_{k-1}$ and largest anticlique of $G^{(0)}_{k-1}$ , respectively. Observe that for every $x_1, x_2 \in A$ , it holds that ${\textrm {NAE}}^{k-1}(x_1 \land x_2) = 1$ (whether or not $x_1 = x_2$ ). Likewise it holds that ${\textrm {NAE}}^{k-1}$ $(y_1 \land y_2) = 0$ for every $y_1, y_2 \in B$ . Therefore, the sets

\begin{equation*} B' {\,:\!=\,} A \times A \times A \subset \{0,1\}^{3^k}, \text{ and } A' {\,:\!=\,} \{0,1\}^{3^{k-1}} \times A \times B \subset \{0,1\}^{3^k} \end{equation*}

are an anticlique of $G^{(0)}_k$ , and a clique of $G^{(1)}_k$ , respectively.

Combining the two bounds, it holds that $\omega _k \geqslant 2^{3^{k-1}} \alpha _{k-1} \omega _{k-1} \geqslant 2^{3^{k-1}} \omega _{k-1} \omega _{k-2}^3$ . Taking logs and denoting $a_k {\,:\!=\,} \log (\omega _k)$ , we thus arrive at the linear recurrence $a_k = 3^{k-1} + a_{k-1} + 3 a_{k-2}$ . Denoting $a_k=3^k(1-\varepsilon _k)$ , the above translates into

\begin{equation*}3^k(1-\varepsilon _k)=3^{k-1}+3^{k-1}(1-\varepsilon _{k-1})+3^{k-1}(1-\varepsilon _{k-2})\end{equation*}

i.e.,

(2) \begin{equation} \varepsilon _k=\frac {\varepsilon _{k-1}+\varepsilon _{k-2}}{3} \end{equation}

and we conclude that $a_k=3^k (1 - o_k(1))$ , where the little-oh term is exponentially small in $k$ . To see this consider $\lambda ^2-{\frac {\lambda +1}{3}}$ , the characteristic polynomial of Equation 2, the roots of which are $\frac {1 \pm \sqrt {13}}{6}$ .

8. Open problems

Many intriguing questions regarding the Rank-Ramsey problem remain unanswered. For starters, the growth rate of the function $\nu _d(n)$ is mostly unknown. For bounded $d$ , we have shown (see Section 4) a polynomial separation between $\nu _d(n)$ and $n$ , starting at $d=41$ . Is this true of all $d$ ? Do there exist triangle-free Rank-Ramsey graphs? Concretely,

The paper [Reference Linial, Mendelson, Schechtman and Shraibman42] advocates the perspective that rank is a complexity measure of sign matrices and mentions some additional measures of similar nature like $\gamma _2$ , margin complexity and more. Rather than demand that the complement rank be small, one can similarly investigate graphs with low clique number for which $\gamma _2(\overline {G})$ is also small etc.Footnote ²⁰

We saw several connections between Rank-Ramsey numbers and various graph parameters. Our list is far from exhausting all possible interesting ties. Could there be any relation with the Colin de Verdière parameter [Reference De Verdière20])? With other orthonormal representations of graphs (e.g., see [Reference Laurent and Varvitsiotis48])?

In Section 6, we briefly consider minimum semidefinite rank, and show $n$ -vertex triangle-free graphs $G$ and $H$ , with both $\left (\textrm {msr}(G) - {\textrm {rank}}(A_G + I)\right ) = \Omega (n)$ and with $\left ({\textrm {rank}}(A_H + I) - \textrm {msr}(H)\right ) = \Omega (n)$ . The relation between the two is therefore likely nuanced. We find it interesting to understand what is the least distance, under the rank metric, between $A_G + I$ and a PSD representation matrix $M$ for $G$ . Another quantity which we consider in Section 6, originating in orthonormal representations, is the Lovász number. We give some evidence that the Lovász number bounds the complement rank from below, perhaps up to a multiplicative constant. Thus, we ask:

The regime of unbounded $d$ , which we explore in Section 5 is also of great interest. Recall that we construct a graph of logarithmic clique number, and polynomial rank. This is in stark contrast to the classical Ramsey problem, where in almost all graphs both the clique number and independence number are logarithmic. Naturally, we think that maintaining low-rank should be much harder than controlling the independence number. It is therefore quite natural to ask,

In Subsection 2.2 we begin an exploration of the Rank-Ramsey numbers. Unlike the usual Ramsey numbers, which are symmetric (i.e., $R(s,t) = R(t,s)$ for every $s$ and $t$ ), this does not hold for our numbers. In fact, we determine the numbers $R^k(s,t)$ for every $2 \leqslant t \leqslant 5$ , and prove that $R^k(3,n) \gt R^k(n,3)$ for sufficiently large $n$ . It would be interesting to better understand the interplay between the clique number and complement rank.

Another perspective of Rank-Ramsey numbers, stems from twin-free graphs. Recall that two vertices in a graph are called twins if they are non-adjacent and have the same set of neighbours. Pruning twins from a graph clearly affects neither the rank and nor chromatic number. Therefore, the sets $\mathcal{G}_r$ of all twin-free connected graphs of rank $r$ , play a crucial role in understanding the log-rank conjecture. Indeed, the log-rank problem (in particular, its graph-theoretic formulation, see Section 3) can be equivalently re-formulated as follows:

Equivalent formulation of Log-Rank problem

Is there a constant $c \gt 0$ such that $ \displaystyle \log \chi (G) \leqslant \mathcal{O} \left ( \log ^c r \right ) \,\text{for every}\,G \in \mathcal{G}_r\,\text{and every} \, r \gt 1 ?$

It would therefore be interesting to investigate the sets $\mathcal{G}_r$ , for a larger range of values $r$ .