A k-uniform family of subsets of [n] is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erdős-Ko-Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when γn ≤ k ≤ ( 1 2 − γ)n, a set family with few disjoint pairs can be made intersecting by removing few sets.We provide a simple proof of a removal lemma for large families, showing that families of size close to ℓ n−1 k−1 with relatively few disjoint pairs must be close to a union of ℓ stars. Our lemma holds for a wide range of uniformities; in particular, when ℓ = 1, the result holds for all 2 ≤ k < n 2 and provides sharp quantitative estimates. We use this removal lemma to answer a question of Bollobás, Narayanan and Raigorodskii regarding the independence number of random subgraphs of the Kneser graph K(n, k). The Erdős-Ko-Rado theorem shows α(K(n, k)) = n−1 k−1 . For some constant c > 0 and k ≤ cn, we determine the sharp threshold for when this equality holds for random subgraphs of K(n, k), and provide strong bounds on the critical probability for k ≤ 1 2 (n − 3).

Consider integers k,ℓ such that 0⩽ℓ⩽0ptk2. Given a large graph G, what is the fraction of k‐vertex subsets of G which span exactly ℓ edges? When G is empty or complete, and ℓ is zero or ()k2, this fraction can be exactly 1. On the other hand, if ℓ is far from these extreme values, one might expect that this fraction is substantially smaller than 1. This was recently proved by Alon, Hefetz, Krivelevich, and Tyomkyn who initiated the systematic study of this question and proposed several natural conjectures. Let ℓ∗=minfalse{ℓ,()k2−ℓfalse}. Our main result is that for any k and ℓ, the fraction of k‐vertex subsets that span ℓ edges is at most prefixlogO(1)false(ℓ∗/kfalse)k/ℓ∗, which is best‐possible up to the logarithmic factor. This improves on multiple results of Alon, Hefetz, Krivelevich, and Tyomkyn, and resolves one of their conjectures. In addition, we also make some first steps towards some analogous questions for hypergraphs. Our proofs involve some Ramsey‐type arguments, and a number of different probabilistic tools, such as polynomial anticoncentration inequalities, hypercontractivity, and a coupling trick for random variables defined on a ‘slice’ of the Boolean hypercube.

The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least c H log d/ log log d, thus nearly confirming one and proving another conjecture of Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdős, Janson,

A set of integers is called sum-free if it contains no triple (x, y, z) of not necessarily distinct elements with x+y = z. In this paper, we provide a structural characterisation of sum-free subsets of {1, 2, . . . , n} of density at least 2/5 − c, where c is an absolute positive constant. As an application, we derive a stability version of Hu's Theorem [Proc. Amer. Math. Soc. 80 (1980), 711-712] about the maximum size of a union of two sum-free sets in {1, 2, . . . , n}. We then use this result to show that the number of subsets of {1, 2, . . . , n} which can be partitioned into two sum-free sets is Θ(2 4n/5 ), confirming a conjecture of Hancock, Staden and Treglown [arXiv:1701.04754].

The classical Erdős-Szekeres theorem dating back almost a hundred years states that any sequence of (n − 1) 2 + 1 distinct real numbers contains a monotone subsequence of length n. This theorem has been generalised to higher dimensions in a variety of ways but perhaps the most natural one was proposed by Fishburn and Graham more than 25 years ago. They defined the concept of a monotone and a lex-monotone array and asked how large an array one needs in order to be able to find a monotone or a lex-monotone subarray of size n × . . . × n. Fishburn and Graham obtained Ackerman-type bounds in both cases. We significantly improve these results. Regardless of the dimension we obtain at most a triple exponential bound in n in the monotone case and a quadruple exponential one in the lex-monotone case. Definition (Monotone array). A d-dimensional arrayone of the following alternatives occurs:(i) f (a 1 , . . . , a i−1 , x, a i+1 , . . . , a d ) is increasing in x for all choices of a 1 , . . . , a i−1 , a i+1 , . . . , a d ;