Triangles in r-wise t-intersecting families

TL;DR

This paper characterizes the structure of large r-wise t-intersecting families that maximize the number of (r+1,t)-triangles, extending Turán-type results in combinatorics.

Contribution

It proves the extremal structure of r-wise t-intersecting families maximizing (r+1,t)-triangles, generalizing previous combinatorial theorems.

Findings

Identifies the family structure that maximizes (r+1,t)-triangles.

Provides a threshold n_0(r, t, k) for the main result.

Extends Turán-type extremal combinatorics results.

Abstract

Let $t$, $r$, $k$ and $n$ be positive integers and $\mathcal{F}$ a family of $k$-subsets of an $n$-set $V$. The family $ \CF $ is $ r $-wise $ t $-intersecting if for any $ F_1, \ldots, F_r \in \CF $, we have $ \abs{\cap_{i = 1}^{r}F_i}\gs t $. An $ r $-wise $ t $-intersecting family of $ r + 1 $ sets $ \{T_1, \ldots, T_{r + 1}\} $ is called an $ (r + 1,t) $-triangle if $ |T_1 \cap \cdots \cap T_{r + 1}| \ls t - 1 $. In this paper, we prove that if $ n \gs n_0(r, t, k) $, then the $ r $-wise $ t $-intersecting family $ \CF \subseteq \binom{[n]}{k} $ containing the most $ (r + 1,t) $-triangles is isomorphic to $ \curlybraces{F \in \binom{[n]}{k}: \abs{F \cap [r + t]} \gs r + t - 1} $. This can also be regarded as a generalized Tur\'{a}n type result.