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Fifty years ago Erd˝ os asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l − 1 complete graphs of size n l−1. This conjecture was disproved by Nikiforov who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques… (More)

The study of intersecting structures is central to extremal combinatorics. A family of permutations F ⊂ S n is t-intersecting if any two permutations in F agree on some t indices, and is trivial if all permutations in F agree on the same t indices. A k-uniform hypergraph is t-intersecting if any two of its edges have t vertices in common, and trivial if all… (More)

The celebrated Erd˝ os-Ko-Rado theorem shows that for n 2k the largest intersecting k-uniform set family on [n] has size n−1 k−1. It is natural to ask how far from intersecting larger set families must be. Katona, Katona and Katona introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being… (More)

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family F of subsets of [n] with all pairwise intersections of size λ can have at most n non-empty sets. One may weaken the condition by requiring that for every set in F, all but at most k of its pairwise intersections have size λ. We call such families k-almost… (More)

An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph H, the rainbow Turán number ex * (n, H) is defined as the maximum number of edges in a properly edge-colored graph on n vertices with no rainbow copy of H. We study the rainbow Turán number of even cycles, and prove that for every fixed ε > 0, there is a… (More)

Given a family F of subsets of [n], we say two sets A, B ∈ F are comparable if A ⊂ B or B ⊂ A. Sperner's celebrated theorem gives the size of the largest family without any comparable pairs. This result was later generalised by Kleitman, who gave the minimum number of comparable pairs appearing in families of a given size. In this paper we study a… (More)

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 Erd˝ os–Ko–Rado theorem of 1961 that bounds the largest such families. A recent trend has been to investigate the structure of set families with few disjoint… (More)

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