On the sizes of t-intersecting k-chain-free families

Balogh, J. and Linz, W.B. and Patkós, Balázs (2023) On the sizes of t-intersecting k-chain-free families. COMBINATORIAL THEORY, 3 (2). ISSN 2766-1334

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Abstract

A set system F is t-intersecting, if the size of the intersection of every pair of its elements has size at least t. A set system F is k-Sperner, if it does not contain a chain of length k + 1. Our main result is the following: Suppose that k and t are fixed positive integers, where n + t is even with t ≤ n and n is large enough. If F ⊆ 2[n] is a t-intersecting k-Sperner family, then |F| has size at most the size of the sum of k layers, of sizes (n + t)/2, . . . , (n + t)/2 + k − 1. This bound is best possible. The case when n + t is odd remains open.

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