A meta-conjecture of Coulson, Keevash, Perarnau, and Yepremyan [12] states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle.

Fix integers $r \ge 2$ and $1\le s_1\le \cdots \le s_{r-1}\le t$ and set $s=\prod _{i=1}^{r-1}s_i$. Let $K=K(s_1, \ldots , s_{r-1}, t)$ denote the complete $r$-partite $r$-uniform hypergraph with parts of size $s_1, \ldots , s_{r-1}, t$. We prove that the Zarankiewicz number $z(n, K)= n^{r-1/s-o(1)}$ provided $t\gt 3^{s+o(s)}$. Previously this was known only for $t \gt ((r-1)(s-1))!$ due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of $s_i$, for example, it gives $z(n, K(2,2,7))=n^{11/4-o(1)}$ where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.

Asymptotic dimension and Assouad–Nagata dimension are measures of the large-scale shape of a class of graphs. Bonamy, Bousquet, Esperet, Groenland, Liu, Pirot, and Scott [J. Eur. Math. Society] showed that any proper minor-closed class has asymptotic dimension 2, dropping to 1 only if the treewidth is bounded. We improve this result by showing it also holds for the stricter Assouad–Nagata dimension. We also characterise when subdivision-closed classes of graphs have bounded Assouad–Nagata dimension.

It is well known that almost all graphs are canonizable by a simple combinatorial routine known as colour refinement, also referred to as the 1-dimensional Weisfeiler–Leman algorithm. With high probability, this method assigns a unique label to each vertex of a random input graph and, hence, it is applicable only to asymmetric graphs. The strength of combinatorial refinement techniques becomes a subtle issue if the input graphs are highly symmetric. We prove that the combination of colour refinement and vertex individualization yields a canonical labelling for almost all circulant digraphs (i.e., Cayley digraphs of a cyclic group). This result provides first evidence of good average-case performance of combinatorial refinement within the class of vertex-transitive graphs. Remarkably, we do not even need the full power of the colour refinement algorithm. We show that the canonical label of a vertex $v$ can be obtained just by counting walks of each length from $v$ to an individualized vertex. Our analysis also implies that almost all circulant graphs are compact in the sense of Tinhofer, that is, their polytops of fractional automorphisms are integral. Finally, we show that a canonical Cayley representation can be constructed for almost all circulant graphs by the more powerful 2-dimensional Weisfeiler–Leman algorithm.

A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov’s result in the following form. Given $r,t\in \mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include:

• • For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline {G}$ has positive triangle density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline {G}$ has positive $K_r$-density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2.

• • Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $\Omega ({n \over {\log n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $\Omega (n)$, where the constant $(2h-2)^{r}+1$ is optimal.

The ${n \over {\log n}}$ size of the blowups in all our results are optimal up to a constant factor.

Given $n$ convex bodies in the Euclidean space $\mathbb{R}^d$, we can find their volume polynomial which is a homogeneous polynomial of degree $d$ in $n$ variables. We consider the set of homogeneous polynomials of degree $d$ in $n$ variables that can be represented as the volume polynomial of any such given convex bodies. This set is a subset of the set of Lorentzian polynomials. Using our knowledge of operations that preserve the Lorentzian property, we give a complete classification of the cases for $(n,d)$ when the two sets are equal.

Balister, the second author, Groenland, Johnston, and Scott recently showed that there are asymptotically $C4^n/n^{3/4}$ many unordered sequences that occur as degree sequences of graphs with $n$ vertices. Combining limit theory for infinitely divisible distributions with a new connection between a class of random walk trajectories and a subset counting formula from additive number theory, we describe $C$ in terms of Walkup’s number of rooted plane trees. The bijection is related to an instance of the Lévy–Khintchine formula. Our main result complements a result of Stanley, that ordered graphical sequences are related to quasi-forests.

Let $K^r_n$ be the complete $r$-uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n] \, :\! = \{1,2,\ldots ,n\}$ and whose edge set is $\binom {[n]}{r}$. We form $G^r(n,p)$ by retaining each edge of $K^r_n$ independently with probability $p$. An $r$-uniform hypergraph $H\subseteq G$ is $F$-saturated if $H$ does not contain any copy of $F$, but any missing edge of $H$ in $G$ creates a copy of $F$. Furthermore, we say that $H$ is weakly $F$-saturated in $G$ if $H$ does not contain any copy of $F$, but the missing edges of $H$ in $G$ can be added back one-by-one, in some order, such that every edge creates a new copy of $F$. The smallest number of edges in an $F$-saturated hypergraph in $G$ is denoted by ${\textit {sat}}(G,F)$, and in a weakly $F$-saturated hypergraph in $G$ by $\mathop {\mbox{$w$-${sat}$}}\! (G,F)$. In 2017, Korándi and Sudakov initiated the study of saturation in random graphs, showing that for constant $p$, with high probability ${\textit {sat}}(G(n,p),K_s)=(1+o(1))n\log _{\frac {1}{1-p}}n$, and $\mathop {\mbox{$w$-${sat}$}}\! (G(n,p),K_s)=\mathop {\mbox{$w$-${sat}$}}\! (K_n,K_s)$. Generalising their results, in this paper, we solve the saturation problem for random hypergraphs $G^r(n,p)$ for cliques $K_s^r$, for every $2\le r \lt s$ and constant $p$.

In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs whose smallest eigenvalue is at least $-2$ by relating such graphs to root systems that appear in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs whose smallest eigenvalue lies in $(\! -\lambda ^*, -2)$, where $\lambda ^* = ho ^{1/2} + ho ^{-1/2} \approx 2.01980$, and $ho$ is the unique real root of $x^3 = x + 1$. Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in $(\! -\lambda , -2)$ for any constant $\lambda \gt 2$.

Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H}$ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all vertices in a particular subset of $ V(\mathcal{H})$, by using an additional set of edges; in particular, we ensure that our matching avoids all additional conflicts, which may consist of both old and new edges. This setup is useful for various applications in design theory and Ramsey theory. For example, our main result provides a crucial tool in the recent proof of the high-girth existence conjecture due to Delcourt and Postle. It also provides a black box which encapsulates many long and tedious calculations, greatly simplifying the proofs of results in generalised Ramsey theory.

We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving normal subgroups of quotients of the group, we show that the random walk converges to the uniform distribution on the group and give bounds for the convergence rate using spectral properties of the random walk steps. As an application, we use the moment method of Wood to prove a universality theorem for cokernels of random integer matrices allowing some dependence between entries.

Here we consider the hypergraph Turán problem in uniformly dense hypergraphs as was suggested by Erdős and Sós. Given a $3$-graph $F$, the uniform Turán density $\pi _{\boldsymbol{\therefore }}(F)$ of $F$ is defined as the supremum over all $d\in [0,1]$ for which there is an $F$-free uniformly $d$-dense $3$-graph, where uniformly $d$-dense means that every linearly sized subhypergraph has density at least $d$. Recently, Glebov, Král’, and Volec and, independently, Reiher, Rödl, and Schacht proved that $\pi _{\boldsymbol{\therefore }}(K_4^{(3)-})=\frac {1}{4}$, solving a conjecture by Erdős and Sós. Despite substantial attention, the uniform Turán density is still only known for very few hypergraphs. In particular, the problem due to Erdős and Sós to determine $\pi _{\boldsymbol{\therefore }}(K_4^{(3)})$ remains wide open.

In this work, we determine the uniform Turán density of the $3$-graph on five vertices that is obtained from $K_4^{(3)-}$ by adding an additional vertex whose link forms a matching on the vertices of $K_4^{(3)-}$. Further, we point to two natural intermediate problems on the way to determining $\pi _{\boldsymbol{\therefore }}(K_4^{(3)})$, and solve the first of these.

A trace of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied trace reconstruction problem asks how many traces are required to reconstruct an unknown binary sequence with high probability. In this paper, we study the multidimensional version of this problem for matrices and hypermatrices, where a trace is generated by deleting each row/column of the matrix or each slice of the hypermatrix independently with a constant probability. Previously, Krishnamurthy, Mazumdar, McGregor and Pal showed that $\exp (\widetilde {O}(n^{d/(d+2)}))$ traces suffice to reconstruct any unknown $n\times n$ matrix (for $d=2$) and any unknown $n^{\times d}$ hypermatrix. By developing a dimension reduction procedure and establishing a multivariate version of the Littlewood-type result that lower bounds sparse complex polynomials around $1$, we improve this upper bound by showing that $\exp (\widetilde {O}(n^{3/7}))$ traces suffice to reconstruct any unknown $n\times n$ matrix, and $\exp (\widetilde {O}(n^{3/5}))$ traces suffice to reconstruct any unknown $n^{\times d}$ hypermatrix. In contrast to the earlier bound, our new exponent is bounded away from $1$ even as $d$ becomes very large.

A seminal result of Komlós, Sárközy, and Szemerédi states that any $n$-vertex graph $G$ with minimum degree at least $(1/2+\alpha )n$ contains every $n$-vertex tree $T$ of bounded degree. Recently, Pham, Sah, Sawhney, and Simkin extended this result to show that such graphs $G$ in fact support an optimally spread distribution on copies of a given $T$, which implies, using the recent breakthroughs on the Kahn-Kalai conjecture, the robustness result that $T$ is a subgraph of sparse random subgraphs of $G$ as well. Pham, Sah, Sawhney, and Simkin construct their optimally spread distribution by following closely the original proof of the Komlós-Sárközy-Szemerédi theorem which uses the blow-up lemma and the Szemerédi regularity lemma. We give an alternative, regularity-free construction that instead uses the Komlós-Sárközy-Szemerédi theorem (which has a regularity-free proof due to Kathapurkar and Montgomery) as a black box. Our proof is based on the simple and general insight that, if $G$ has linear minimum degree, almost all constant-sized subgraphs of $G$ inherit the same minimum degree condition that $G$ has.

We show that for any integer $k\ge 1$ there exists an integer $t_0(k)$ such that, for integers $t, k_1, \ldots , k_{t+1}, n$ with $t\gt t_0(k)$, $\max \{k_1, \ldots , k_{t+1}\}\le k$, and $n \gt 2k(t+1)$, the following holds: If $F_i$ is a $k_i$-uniform hypergraph with vertex set $[n]$ and more than $ \binom{n}{k_i}-\binom{n-t}{k_i} - \binom{n-t-k}{k_i-1} + 1$ edges for all $i \in [t+1]$, then either $\{F_1,\ldots , F_{t+1}\}$ admits a rainbow matching of size $t+1$ or there exists $W\in \binom{[n]}{t}$ such that $W$ intersects $F_i$ for all $i\in [t+1]$. This may be viewed as a rainbow non-uniform extension of the classical Hilton-Milner theorem. We also show that the same holds for every $t$ and $n \gt 2k^3t$, generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings.

The famous Sidorenko’s conjecture asserts that for every bipartite graph $H$, the number of homomorphisms from $H$ to a graph $G$ with given edge density is minimised when $G$ is pseudorandom. We prove that for any graph $H$, a graph obtained from replacing edges of $H$ by generalised theta graphs consisting of even paths satisfies Sidorenko’s conjecture, provided a certain divisibility condition on the number of paths. To achieve this, we prove unconditionally that bipartite graphs obtained from replacing each edge of a complete graph with a generalised theta graph satisfy Sidorenko’s conjecture, which extends a result of Conlon, Kim, Lee and Lee [J. Lond. Math. Soc., 2018].

In this article, we study a non-uniform distribution on permutations biased by their number of records that we call record-biased permutations. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterise. This model was introduced in our earlier work [3], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.

We prove that determining the weak saturation number of a host graph $F$ with respect to a pattern graph $H$ is computationally hard, even when $H$ is the triangle. Our main tool establishes a connection between weak saturation and the shellability of simplicial complexes.

In the 1980s, Erdős and Sós initiated the study of Turán problems with a uniformity condition on the distribution of edges: the uniform Turán density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least $d$ contains $H$. In particular, they asked to determine the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349 – 366] and [J. Eur. Math. Soc. 20 (2018), 1139 – 1159], while the latter still remains open. Till today, there are known constructions of $3$-uniform hypergraphs with uniform Turán density equal to $0$, $1/27$, $4/27$, and $1/4$ only. We extend this list by a fifth value: we prove an easy to verify sufficient condition for the uniform Turán density to be equal to $8/27$ and identify hypergraphs satisfying this condition.

For $\ell \geq 3$, an $\ell$-uniform hypergraph is disperse if the number of edges induced by any set of $\ell +1$ vertices is 0, 1, $\ell$, or $\ell +1$. We show that every disperse $\ell$-uniform hypergraph on $n$ vertices contains a clique or independent set of size $n^{\Omega _{\ell }(1)}$, answering a question of the first author and Tomon. To this end, we prove several structural properties of disperse hypergraphs.