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.

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.

In this article, we present a unified approach for proving several Turán-type and generalized Turán-type problems, degree power problems, and extremal spectra problems on paths, cycles, and matchings. Specifically, we generalize classical results on cycles and matchings by Kopylov, Erdős–Gallai, and Luo et al., respectively, and provide a positive resolution to an open problem originally proposed by Nikiforov. Moreover, we improve the spectral extremal results on paths, building on the work of Nikiforov, and Nikiforov and Yuan. Additionally, we provide a comprehensive solution to the connected version of the problem related to the degree power sum of a graph that contains no path on k vertices, a topic initially investigated by Caro and Yuster.

The Pósa–Seymour conjecture determines the minimum degree threshold for forcing the $k$th power of a Hamilton cycle in a graph. After numerous partial results, Komlós, Sárközy, and Szemerédi proved the conjecture for sufficiently large graphs. In this paper, we focus on the analogous problem for digraphs and for oriented graphs. We asymptotically determine the minimum total degree threshold for forcing the square of a Hamilton cycle in a digraph. We also give a conjecture on the corresponding threshold for $k$th powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the $k$th power of a Hamilton cycle; although this minimum semi-degree condition is not tight, it does provide the correct order of magnitude of the threshold. Turán-type problems for oriented graphs are also discussed.

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.

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.

We prove that there is an absolute constant $C{\,\gt\,}0$ such that every k-vertex connected rainbow graph R with minimum degree at least $C\log k$ has inducibility $k!/(k^k-k)$. The same result holds if $k\ge 11$, and R is a clique. This answers a question posed by Huang, that is a generalisation of an old problem of Erdös and Sós. It remains open to determine the minimum k for which this is true.

For a family $\mathcal {F}$ of graphs, let ${\mathrm {ex}}(n,\mathcal {F})$ denote the maximum number of edges in an n-vertex graph which contains none of the members of $\mathcal {F}$ as a subgraph. A longstanding problem in extremal graph theory asks to determine the function ${\mathrm {ex}}(n,\{C_3,C_4\})$. Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of ${\mathrm {ex}}(n,\{C_3,C_4\})$ since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3].

A simple k-coloring of a multigraph G is a decomposition of the edge multiset as a disjoint sum of k simple graphs which are referred to as colors. A subgraph H of a multigraph G is called multicolored if its edges receive distinct colors in a given simple k-coloring of G. In 2004, Keevash–Saks–Sudakov–Verstraëte introduced the k-color Turán number ${\text {ex}}_k(n,H)$, which denotes the maximum number of edges in an n-vertex multigraph that has a simple k-coloring containing no multicolored copies of H. They made a conjecture for any $r\geq 3$ and r-color-critical graph $H,$ that in the range of $k\geq \frac {r-1}{r-2}(e(H)-1)$, if n is sufficiently large, then ${\text {ex}}_k(n, H)$ is achieved by the multigraph consisting of k colors all of which are identical copies of the Turán graph $T_{r-1}(n)$. In this article, we show that this holds in the range of $k\geq 2\frac {r-1}{r}(e(H)-1)$, significantly improving earlier results. Our proof combines the stability argument of Chakraborti–Kim–Lee–Liu–Seo with a novel graph packing technique for embedding multigraphs.

The Erdős-Sós Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta \gt 0$ and $k_0\in \mathbb N$ such that the conjecture holds for every tree $T$ with $k \ge k_0$ edges and every graph $G$ with $|V(G)| \le (1+\delta )|V(T)|$.

The Erdős–Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős–Simonovits type stability theorem in multi-partite graphs. Different from the Erdős–Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an $H$-free graph $G$ is close to the extremal graphs for $H$, then $G$ has a well-defined structure but may be far away from the extremal graphs for $H$. As applications, we strengthen a theorem of Bollobás, Erdős, and Straus and solve a conjecture in a stronger form posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique.

We consider the hypergraph Turán problem of determining $ex(n, S^d)$, the maximum number of facets in a $d$-dimensional simplicial complex on $n$ vertices that does not contain a simplicial $d$-sphere (a homeomorph of $S^d$) as a subcomplex. We show that if there is an affirmative answer to a question of Gromov about sphere enumeration in high dimensions, then $ex(n, S^d) \geq \Omega (n^{d + 1 - (d + 1)/(2^{d + 1} - 2)})$. Furthermore, this lower bound holds unconditionally for 2-LC (locally constructible) spheres, which includes all shellable spheres and therefore all polytopes. We also prove an upper bound on $ex(n, S^d)$ of $O(n^{d + 1 - 1/2^{d - 1}})$ using a simple induction argument. We conjecture that the upper bound can be improved to match the conditional lower bound.

A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimised by a random colouring in which each edge is equally likely to be red or blue. We extend this notion to an off-diagonal setting. That is, we define a pair $(H_1,H_2)$ of graphs to be $(p,1-p)$-common if a particular linear combination of the density of $H_1$ in red and $H_2$ in blue is asymptotically minimised by a random colouring in which each edge is coloured red with probability $p$ and blue with probability $1-p$. Our results include off-diagonal extensions of several standard theorems on common graphs and novel results for common pairs of graphs with no natural analogue in the classical setting.

In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$ with minimum $(k-1)$-degree $\delta (G) \ge (1/2+o(1))n$ contains a tight Hamilton cycle with high discrepancy, that is, with at least $n/r+\Omega (n)$ edges of one colour. The minimum degree condition is asymptotically best possible and our theorem also implies a corresponding result for perfect matchings. Our tools combine various structural techniques such as Turán-type problems and hypergraph shadows with probabilistic techniques such as random walks and the nibble method. We also propose several intriguing problems for future research.

We present a short and simple proof of the celebrated hypergraph container theorem of Balogh–Morris–Samotij and Saxton–Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree, which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.

A simple probabilistic argument shows that every r-uniform hypergraph with m edges contains an r-partite subhypergraph with at least $({r!}/{r^r})m$ edges. The celebrated result of Edwards states that in the case of graphs, that is $r=2$, the resulting bound $m/2$ can be improved to $m/2+\Omega(m^{1/2})$, and this is sharp. We prove that if $r\geq 3$, then there is an r-partite subhypergraph with at least $({r!}/{r^r}) m+m^{3/5-o(1)}$ edges. Moreover, if the hypergraph is linear, this can be improved to $({r!}/{r^r}) m+m^{3/4-o(1)}$, which is tight up to the o(1) term. These improve results of Conlon, Fox, Kwan and Sudakov. Our proof is based on a combination of probabilistic, combinatorial, and linear algebraic techniques, and semidefinite programming.

A key part of our argument is relating the energy $\mathcal{E}(G)$ of a graph G (i.e. the sum of absolute values of eigenvalues of the adjacency matrix) to its maximum cut. We prove that every m edge multigraph G has a cut of size at least $m/2+\Omega({\mathcal{E}(G)}/{\log m})$, which might be of independent interest.

Given a fixed k-uniform hypergraph F, the F-removal lemma states that every hypergraph with few copies of F can be made F-free by the removal of few edges. Unfortunately, for general F, the constants involved are given by incredibly fast-growing Ackermann-type functions. It is thus natural to ask for which F one can prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when F is k-partite. Alon proved that when $k=2$ (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and Rödl conjectured in 2002 that Alon’s result can be extended to all $k\gt2$, namely, that the only $k$-graphs $F$ for which the hypergraph removal lemma has polynomial bounds are the trivial cases when F is k-partite. In this paper we prove this conjecture.

A conjecture of Jackson from 1981 states that every d-regular oriented graph on n vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large n. In fact we prove a more general result that for all $\alpha>0$, there exists $n_0=n_0(\alpha )$ such that every d-regular digraph on $n\geq n_0$ vertices with $d \ge \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles, and moreover that if G is an oriented graph, then at most $n/(2d+1)$ cycles suffice.

We study the generalized Ramsey–Turán function $\mathrm {RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an n-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by Erdős, Sós, Bollobás, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all $s\ge 3$, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when t is much larger than s. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold.

A fundamental extremality result due to Sidorenko [‘A partially ordered set of functionals corresponding to graphs’, Discrete Math. 131(1–3) (1994), 263–277] states that among all connected graphs G on k vertices, the k-vertex star maximises the number of graph homomorphisms of G into any graph H. We provide a new short proof of this result using only a simple recursive counting argument for trees and Hölder’s inequality.