The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families

2024/02/26 by Sumin Huang, Huang, Sumin
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2402.16730

Abstract

Let ω(F)=∑_\A,B\\subsetF|A∩ B| and ω(A,B)=∑_{(A,B)∈ A× B}|A∩ B|. A family F is intersecting if F₁∩ F₂≠ ∅ for any F₁,F₂\inF and two family A and B are crossing-intersecting if A∩ B≠ ∅ for any (A,B)∈ A\timesB. For an intersecting family F, Erdős, Ko and Rado determined the upper bound of |F|, consequently yielding an upper bound of \binom|F|2=∑_\A,B\\subsetF1. If we replace 1 with |A∩ B| in the summation ∑_\A,B\\subsetF1, then this summation transforms into ω(F). In this paper, for an intersecting family F, we determine the upper bound of ω(F), which is a generalization of Erdős-Ko-Rado Theorem. Further, for crossing-intersecting families A and B, we determine the upper bound of ω(A,B).

Related

• The maximum sum of sizes of non-empty cross t-intersecting families
• The maximum sum and maximum product of sizes of cross-intersecting\n families
• The maximum sum of sizes of non-empty cross L-intersecting families
• The maximum product of sizes of cross-\(t\)-intersecting families
• The maximum sum of sizes of cross-intersecting families of subsets of a set