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This article is available for purchase or by subscription. See below.
Extremal problems concerning ordered intersections of sets

Peter Frankl and Jian Wang

Vol. 13 (2024), No. 2, 123–140
Abstract

For two subsets A = {a1,,ak} and B = {b1,,bk} of [n] with a1 < a2 < < ak and b1 < b2 < < bk, we define their coincidence by |A B| = |{i : ai = bi}|. A family [n] k is called t-coinciding if |F F| t for all distinct F,F. It is conjectured that for n k t > 0 a t-coinciding family has size at most nt kt. The conjecture is confirmed for cases t = 1,k 1,k 2, the case n = k + 1, and the case n = k + 2, k 3t 1. The opposite problem, where k-subsets are forbidden to coincide in t or more elements leads to new types of designs. We solve the new design problem completely for the t = 2 case.

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Keywords
extremal set theory, ordered intersection, $t$-coinciding
Mathematical Subject Classification
Primary: 05D05
Secondary: 05B30
Milestones
Received: 11 October 2023
Revised: 18 March 2024
Accepted: 3 April 2024
Published: 16 May 2024
Authors
Peter Frankl
Rényi Institute
Hungarian Academy of Sciences
Budapest
Hungary
Jian Wang
Department of Mathematics
Taiyuan University of Technology
Taiyuan
China