On the maximum number of distinct intersections in an intersecting family

TL;DR

This paper investigates the maximum number of distinct intersections in intersecting families of k-subsets of an n-set, proving that a specific family maximizes this number when n is sufficiently large.

Contribution

It establishes that for large n, the family of k-sets intersecting {1,2,3} in at least two elements maximizes the number of distinct intersections.

Findings

The maximum number of distinct intersections is achieved by the family .

The family  is optimal for n  50k^2.

The result holds for sufficiently large n, specifically n  50k^2.

Abstract

For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in \mathcal F$. Let $\mathcal A$ consist of the $k$-sets $A$ satisfying $|A \cap \{1, 2, 3\}| \geq 2$. We prove that for $n \geq 50 k^2$ $|\mathcal I(\mathcal F)|$ is maximized by $\mathcal A$.