Triangles in intersecting families

Dániel T. Nagy; Balázs Patkós

paper

Triangles in intersecting families

arXiv:2201.02452

Abstract

We prove the following the generalized Turán type result. A collection $T$ of $r$ sets is an $r$ -triangle if for every $T_{1}, T_{2}, \dots, T_{r - 1} \in T$ we have $\cap_{i = 1}^{r - 1} T_{i} \neq = \emptyset$ , but $\cap_{T \in T} T$ is empty. A family $F$ of sets is $r$ -wise intersecting if for any $F_{1}, F_{2}, \dots, F_{r} \in F$ we have $\cap_{i = 1}^{r} F_{i} \neq = \emptyset$ or equivalently if $F$ does not contain any $m$ -triangle for $m = 2, 3, \dots, r$ . We prove that if $n \geq n_{0} (r, k)$ , then the $r$ -wise intersecting family $F \subseteq (k [ n ])$ containing the most number of $(r + 1)$ -triangles is isomorphic to ${F \in (k [ n ]) : ∣ F \cap [r + 1] ∣ \geq r}$ .

7 pages