Triangles in intersecting families

TL;DR

This paper characterizes the maximum size of certain intersecting set families avoiding specific configurations called r-triangles, extending Turán-type results in combinatorics.

Contribution

It establishes the extremal structure of r-wise intersecting families that maximize (r+1)-triangles, generalizing previous Turán-type theorems.

Findings

Identifies the extremal family structure for large n.

Proves the maximum number of (r+1)-triangles in r-wise intersecting families.

Shows the extremal family is isomorphic to a specific subset family.

Abstract

We prove the following the generalized Tur\'an type result. A collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every $T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have $\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in \mathcal{T}}T$ is empty. A family $\mathcal{F}$ of sets is $r$-wise intersecting if for any $F_1,F_2,\dots,F_r\in \mathcal{F}$ we have $\cap_{i=1}^rF_i\neq \emptyset$ or equivalently if $\mathcal{F}$ does not contain any $m$-triangle for $m=2,3,\dots,r$. We prove that if $n\ge n_0(r,k)$, then the $r$-wise intersecting family $\mathcal{F}\subseteq \binom{[n]}{k}$ containing the most number of $(r+1)$-triangles is isomorphic to $\{F\in \binom{[n]}{k}:|F\cap [r+1]|\ge r\}$.