Shagnik Das

A problem of Erdős on the minimum number of k-cliques

Fifty years ago Erdos 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 nl-1. This conjecture was disproved by Nikiforov, who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size n2. In this paper we solve Erdos@? problem for (k,l)=(3,4) and (k,l)=(4,3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdos@? conjecture for (k,l)=(3,4) and showing that a blow-up of a 5-cycle gives the minimum for (k,l)=(4,3).

Intersecting families of discrete structures are typically trivial

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 its edges share the same t vertices.The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for n sufficiently large with respect to t, the largest t-intersecting families in S n are the trivial ones. The classic Erd?s-Ko-Rado theorem shows that the largest t-intersecting k-uniform hypergraphs are also trivial when n is large. We determine the typical structure of t-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings.Our proofs use the Bollobas set-pairs inequality to bound the number of maximal intersecting families, which can then be combined with known stability theorems. We also obtain similar results for vector spaces.

Sperner's Theorem and a Problem of Erdős, Katona and Kleitman

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. Erdýos extended this theorem to determine the largest family without a k-chain F1 ⊂ F2 ⊂ ... ⊂ Fk. Erdýos and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds. In 1966, Kleitman resolved this question for 2-chains, showing that the number of such chains is minimized by taking sets as close to the middle level as possible. Moreover, he conjectured the extremal families were the same for k-chains, for all k. In this paper, making the first progress on this problem, we verify Kleitman’s conjecture for the families whose size is at most the size of the k + 1 middle levels. We also characterize all extremal configurations.

The minimum number of disjoint pairs in set systems and related problems

Let F be a set system on [n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdős, Ko and Rado from 1961 says that, provided n ≥ 2k, any such system has size at most

A simple removal lemma for large nearly-intersecting families

(_{k - 1}^{n - 1})

Almost-Fisher families

(k−1n−1). A natural question, which was asked by Ahlswede in 1980, is how many disjoint pairs must appear in a set system of larger size. Except for the case k = 2, solved by Ahlswede and Katona, this problem has remained open for the last three decades.In this paper, we determine the minimum number of disjoint pairs in small k-uniform families, thus confirming a conjecture of Bollobás and Leader in these cases. Moreover, we obtain similar results for two well-known extensions of the Erdős-Ko-Rado Theorem, determining the minimum number of matchings of size q and the minimum number of t-disjoint pairs that appear in set systems larger than the corresponding extremal bounds. In the latter case, this provides a partial solution to a problem of Kleitman and West.

The typical structure of intersecting families of discrete structures

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Most Probably Intersecting Hypergraphs

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