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Fifty years ago Erdős 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)
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 largest such families. A recent trend has been to investigate the structure of set families with few disjoint(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 themaximumnumber 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)
A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain F1 ⊂ F2. Erdős extended this theorem to determine the largest family without a k-chain F1 ⊂ F2 ⊂ . . . ⊂ Fk. Erdős and Katona, followed by Kleitman, asked how many chains must appear in(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 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(More)
The study of intersecting structures is central to extremal combinatorics. A family of permutations F ⊂ Sn 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 tintersecting if any two of its edges have t vertices in common, and trivial if all(More)
The celebrated Erdős-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)
Shagnik Das University of California, Los Angeles Let F be a family of subsets of [n] such that all sets have size k and every pair of sets intersect. The celebrated theorem of Erdős-Ko-Rado from 1961 says that when n ≥ 2k, any such family has size at most ( n−1 k−1 ) . A natural question to ask is how many disjoint pairs must appear in a set system of(More)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv EXECUTIVES UMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . xvii . l.INTRODUCTION .. .:. .................................. l-l 1.1 BACKGROUND .................................... l-l 1.2 MARKETTRENDS ..................................... l-2 1.3(More)