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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.

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