The structure of maximal non-trivial d-wise intersecting uniform families with large sizes

TL;DR

This paper refines the structural understanding of the largest maximal non-trivial d-wise intersecting uniform families, identifying the top configurations for various sizes using the Δ-system method.

Contribution

It provides a detailed characterization of the top maximal non-trivial d-wise intersecting families, extending previous theorems to include the third to sixth largest families.

Findings

Characterization of the second largest family structure

Determination of the third and fourth largest family structures

Asymptotic description of the fifth and sixth largest families

Abstract

For a positive integer $d\geq 2$, a family $\mathcal F\subseteq \binom{[n]}{k}$ is said to be d-wise intersecting if $|F_1\cap F_2\cap \dots\cap F_d|\geq 1$ for all $F_1, F_2, \dots ,F_d\in \mathcal F$. A d-wise intersecting family $\mathcal F\subseteq \binom{[n]}{k}$ is called maximal if $\mathcal F\cup\{A\}$ is not d-wise intersecting for any $A\in\binom{[n]}{k}\setminus\mathcal F$. We provide a refinement of O'Neill and Verstra\"{e}te's Theorem about the structure of the largest and the second largest maximal non-trivial d-wise intersecting k-uniform families. We also determine the structure of the third largest and the fourth largest maximal non-trivial d-wise intersecting k-uniform families for any $k>d+1\geq 4$, and the fifth largest and the sixth largest maximal non-trivial 3-wise intersecting k-uniform families for any $k\geq 5$, in the asymptotic sense. Our proofs are applications of the $\Delta$-system method.