Shagnik Das

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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)
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)
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)
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