For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ⩾ 4 and 0 < p < 1. Suppose that H is an n-vertex triple system with r|n and the following two properties:

• • for every graph G with V(G) = V(H), at least p proportion of the triangles in G are also edges of H,

• • for every vertex x of H, the link graph of x is a quasirandom graph with density at least p.

For natural numbers $n,r\in \mathbb{N}$ with $n\geqslant r$, the Kneser graph $K(n,r)$ is the graph on the family of $r$-element subsets of $\{1,\ldots ,n\}$ in which two sets are adjacent if and only if they are disjoint. Delete the edges of $K(n,r)$ with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as $r/n$ is bounded away from $1/2$, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős–Ko–Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.

A set A of positive integers is a Bh-set if all the sums of the form a1 + . . . + ah, with a1,. . .,ah ∈ A and a1 ⩽ . . . ⩽ ah, are distinct. We provide asymptotic bounds for the number of Bh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of Bh-sets. We also use these results to estimate the maximum size of a Bh-sets contained in a typical (random) subset of [n] with a given cardinality.

We show that for any coprime integers λ1, . . ., λk and any finite A ⊂ $\mathbb{Z}$, one has

|\lambda_1 \cdot A + \cdots + \lambda_k \cdot A| \geq (|\lambda_1| + \cdots + |\lambda_k|)|A|- C,

A family of sets is called union-closed if whenever A and B are sets of the family, so is A ∪ B. The long-standing union-closed conjecture states that if a family of subsets of [n] is union-closed, some element appears in at least half the sets of the family. A natural weakening is that the union-closed conjecture holds for large families, that is, families consisting of at least p02n sets for some constant p0. The first result in this direction appears in a recent paper of Balla, Bollobás and Eccles [1], who showed that union-closed families of at least $\tfrac{2}{3}$2n sets satisfy the conjecture; they proved this by determining the minimum possible average size of a set in a union-closed family of given size. However, the methods used in that paper cannot prove a better constant than $\tfrac{2}{3}$. Here, we provide a stability result for the main theorem of [1], and as a consequence we prove the union-closed conjecture for families of at least ($\tfrac{2}{3}$ − c)2n sets, for a positive constant c.

In 1981 Beck and Fiala proved an upper bound for the discrepancy of a set system of degree d that is independent of the size of the ground set. In the intervening years the bound has been decreased from 2d − 2 to 2d − 4. We improve the bound to 2d − log*d.

For two given graphs $G_{1}$ and $G_{2}$, the Ramsey number $R(G_{1},G_{2})$ is the smallest integer $N$ such that, for any graph $G$ of order $N$, either $G$ contains $G_{1}$ as a subgraph or the complement of $G$ contains $G_{2}$ as a subgraph. A fan $F_{l}$ is $l$ triangles sharing exactly one vertex. In this note, it is shown that $R(F_{n},F_{m})=4n+1$ for $n\geq \max \{m^{2}-m/2,11m/2-4\}$.

Dyadic shifts $D\oplus T$ of point distributions $D$ in the $d$-dimensional unit cube $U^{d}$ are considered as a form of randomization. Explicit formulas for the $L_{q}$-discrepancies of such randomized distributions are given in the paper in terms of Rademacher functions. Relying on the statistical independence of Rademacher functions, Khinchin’s inequalities, and other related results, we obtain very sharp upper and lower bounds for the mean $L_{q}$-discrepancies, $0<q\leqslant \infty$. The upper bounds imply directly a generalization of the well-known Chen theorem on mean discrepancies with respect to dyadic shifts (Theorem 2.1). From the lower bounds, it follows that for an arbitrary $N$-point distribution $D_{N}$ and any exponent $0<q\leqslant 1$, there exist dyadic shifts $D_{N}\oplus T$ such that the $L_{q}$-discrepancy ${\mathcal{L}}_{q}[D_{N}\oplus T]>c_{d,q}(\log N)^{(1/2)(d-1)}$ (Theorem 2.2). The lower bounds for the $L_{\infty }$-discrepancy are also considered in the paper. It is shown that for an arbitrary $N$-point distribution $D_{N}$, there exist dyadic shifts $D_{N}\oplus T$ such that ${\mathcal{L}}_{\infty }[D_{N}\oplus T]>c_{d}(\log N)^{(1/2)d}$ (Theorem 2.3).

This paper addresses the following question for a given graph H: What is the minimum number f(H) such that every graph with average degree at least f(H) contains H as a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth when H is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined f(H) when H has a super-linear number of edges. We focus on the case when H has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if H has t vertices and average degree d at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when H has small average degree, we prove that if H has t vertices and q edges, then f(H) ⩽ t + 6.291q (where the coefficient of 1 in the t term is best possible).

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.

Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.

In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

The first open case of the Brown–Erdős–Sós conjecture is equivalent to the following: for every c > 0, there is a threshold n0 such that if a quasigroup has order n ⩾ n0, then for every subset S of triples of the form (a, b, ab) with |S| ⩾ cn2, there is a seven-element subset of the quasigroup which spans at least four triples of S. In this paper we prove the conjecture for finite groups.

The shadow of a system of sets is all sets which can be obtained by taking a set in the original system, and removing a single element. The Kruskal-Katona theorem tells us the minimum possible size of the shadow of $\mathcal A$, if $\mathcal A$ consists of m r-element sets.

In this paper, we ask questions and make conjectures about the minimum possible size of a partial shadow for $\mathcal A$, which contains most sets in the shadow of $\mathcal A$. For example, if $\mathcal B$ is a family of sets containing all but one set in the shadow of each set of $\mathcal A$, how large must $\mathcal B$ be?

In this paper we provide two results. The first one consists of an infinitary version of the Furstenberg–Weiss theorem. More precisely we show that every subset A of a homogeneous tree T such that

$\frac{|A\cap T(n)|}{|T(n)|}\geqslant\delta,$

The second result is the following. For every sequence (mq)q∈ℕ of positive integers and for every real 0 < δ ⩽ 1, there exists a sequence (nq)q∈ℕ of positive integers such that for every D ⊆ ∪k ∏q=0k-1[nq] satisfying

$\frac{\big|D\cap \prod_{q=0}^{k-1} [n_q]\big|s}{\prod_{q=0}^{k-1}n_q}\geqslant\delta$

We study sum-free sets in sparse random subsets of even-order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the case G = ℤ2n, and who obtained a weaker threshold (up to a constant factor) in general.

We provide primitive recursive bounds for the finite version of Gowers’ $c_{0}$ theorem for both the positive and the general case. We also provide multidimensional versions of these results.

In this paper, we consider the so-called “Furstenberg set problem” in high dimensions. First, following Wolff’s work on the two-dimensional real case, we provide “reasonable” upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_{p}$. Next we study the “critical” case and improve the “trivial” exponent by ${\rm\Omega}(1/n^{2})$ for $\mathbb{F}_{p}^{n}$. Our key tool in obtaining this lower bound is a theorem about how things behave when the Loomis–Whitney inequality is nearly sharp, as it helps us to reduce the problem to dimension two.

In this paper we give a short proof of the Random Ramsey Theorem of Rödl and Ruciński: for any graph F which contains a cycle and r ≥ 2, there exist constants c, C > 0 such that

$$\begin{equation*}\Pr[G_{n,p} \rightarrow (F)_r^e] =\begin{cases}1-o(1) &p\ge Cn^{-1/m_2(F)},\\o(1) &p\le cn^{-1/m_2(F)},\end{cases}\end{equation*}$$

$$\begin{equation*}m_2(F) = \max_{J\subseteq F, v_J\ge 2} \frac{e_J-1}{v_J-2}.\end{equation*}$$

We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least

\begin{equation*}\binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}.\end{equation*}

Let ${\mathrm{OT} }_{d} (n)$ be the smallest integer $N$ such that every $N$-element point sequence in ${ \mathbb{R} }^{d} $ in general position contains an order-type homogeneous subset of size $n$, where a set is order-type homogeneous if all $(d+ 1)$-tuples from this set have the same orientation. It is known that a point sequence in ${ \mathbb{R} }^{d} $ that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in ${ \mathbb{R} }^{d} $. Two famous theorems of Erdős and Szekeres from 1935 imply that ${\mathrm{OT} }_{1} (n)= \Theta ({n}^{2} )$ and ${\mathrm{OT} }_{2} (n)= {2}^{\Theta (n)} $. For $d\geq 3$, we give new bounds for ${\mathrm{OT} }_{d} (n)$. In particular, we show that ${\mathrm{OT} }_{3} (n)= {2}^{{2}^{\Theta (n)} } $, answering a question of Eliáš and Matoušek, and, for $d\geq 4$, we show that ${\mathrm{OT} }_{d} (n)$ is bounded above by an exponential tower of height $d$ with $O(n)$ in the topmost exponent.

We generalise an argument of Leader, Russell, and Walters to show that almost all sets of $d+ 2$ points on the $(d- 1)$-sphere ${S}^{d- 1} $ are not contained in a transitive set in some ${\mathbf{R} }^{n} $.