We show that there exist constants $\delta _1,\delta _2\gt 0$ such that if $G$ is an $(n,d,\lambda )$-graph with $\lambda /d\le \delta _1$, then $G$ contains an induced cycle of length at least $\delta _2n/d$. We further demonstrate that, up to a constant factor, this is best possible. Utilising our techniques, we derive that the number of non-isomorphic induced subgraphs of such $G$ is at least exponential in $n\log d/d$, and further demonstrate that this is tight up to a constant factor in the exponent.

Let $\Omega _1, \ldots , \Omega _m$ be probability spaces, let ${\mathbf \Omega }=\Omega _1 \times \cdots \times \Omega _m$ be their product and let $A_1, \ldots , A_n \subset {\mathbf \Omega }$ be events. Suppose that each event $A_i$ depends on $r_i$ coordinates of a point $x \in {\mathbf \Omega }$, $x=\left (\xi _1, \ldots , \xi _m\right )$, and that for each event $A_i$ there are $\Delta _i$ other events $A_j$ that depend on some of the coordinates that $A_i$ depends on. Let $\Delta =\max \{5,\ \Delta _i\,:\, i=1, \ldots , n\}$ and let $\mu _i=\min \{r_i,\ \Delta _i+1\}$ for $i=1, \ldots , n$. We prove that if ${\mathbb P}(A_i) \lt (3\Delta )^{-3\mu _i}$ for all $i$, then for any $0 \lt \epsilon \lt 1$, the probability ${\mathbb P}\left ( \bigcap _{i=1}^n \overline {A}_i\right )$ of the intersection of the complements of all $A_i$ can be computed within relative error $\epsilon$ in polynomial time from the probabilities ${\mathbb P}\left (A_{i_1} \cap \ldots \cap A_{i_k}\right )$ of $k$-wise intersections of the events $A_i$ for $k = e^{O(\Delta )} \ln (n/\epsilon )$.

We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+{\mathrm{o}}(n)$. This can be seen as a directed graph analogue of a well-known theorem of Komlós, Sárközy, and Szemerédi. Our result for trees follows from a more general result, allowing the embedding of arbitrary orientations of a much wider class of spanning ‘tree-like’ structures, such as collections of at most $O(n^{0.99})$ pairwise vertex-disjoint cycles and subdivisions of graphs $H$ with $|H|\lt \exp \bigl (\sqrt {\textrm {O}(\log n)}\,\bigr )$ in which each edge is subdivided at least once.

A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory. We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erdős-Rényi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n^{2/3})$. A key component of this construction is our matrix-theoretic view of lifts. We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey.

We study off-diagonal Ramsey numbers $r(H, K_n^{(k)})$ of $k$-uniform hypergraphs, where $H$ is a fixed linear $k$-uniform hypergraph and $K_n^{(k)}$ is complete on $n$ vertices. Recently, Conlon, Fox, Gunby, He, Mubayi, Suk, and Verstraëte disproved the folklore conjecture that $r(H, K_n^{(3)})$ always grows polynomially in $n$. In this paper, we show that much larger growth rates are possible in higher uniformity. In uniformity $k\ge 4$, we prove that for any constant $C\gt 0$, there exists a linear $k$-uniform hypergraph $H$ for which

\begin{equation*} r(H,K_n^{(k)}) \geq {\textrm {twr}}_{k-2}(2^{(\log n)^C}). \end{equation*}

Confirming a conjecture of Erdős on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lovász proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac {n+q-1}{r+q-1}\rfloor$. In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$, we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$-colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon–Frankl–Lovász theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erdős matching conjecture.

We prove that for any $k\geq 3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$-SAT formulas is given by the ‘replica symmetric solution’ predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. 76 (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $k\geq 3$, we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)]. The improvement is significant particularly for small $k$.

Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$. Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$. In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$. This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$.

Given a collection $\mathcal{D} =\{D_1,D_2,\ldots ,D_m\}$ of digraphs on the common vertex set $V$, an $m$-edge digraph $H$ with vertices in $V$ is transversal in $\mathcal{D}$ if there exists a bijection $\varphi \,:\,E(H)\rightarrow [m]$ such that $e \in E(D_{\varphi (e)})$ for all $e\in E(H)$. Ghouila-Houri proved that any $n$-vertex digraph with minimum semi-degree at least $\frac {n}{2}$ contains a directed Hamilton cycle. In this paper, we provide a transversal generalisation of Ghouila-Houri’s theorem, thereby solving a problem proposed by Chakraborti, Kim, Lee, and Seo. Our proof utilises the absorption method for transversals, the regularity method for digraph collections, as well as the transversal blow-up lemma and the related machinery. As an application, when $n$ is sufficiently large, our result implies the transversal version of Dirac’s theorem, which was proved by Joos and Kim.