A bound for $1$-cross intersecting set pair systems

TL;DR

This paper improves an upper bound on the size of 1-cross intersecting set pair systems by confirming a conjecture that the additional intersection condition allows a constant factor reduction.

Contribution

It proves that the extra intersection condition in 1-cross intersecting systems leads to a tighter upper bound, confirming a conjecture by F"uredi, Gyárfás, and Király.

Findings

Improved upper bound for 1-cross intersecting set pair systems.

Confirmed conjecture that additional intersection condition reduces the bound.

Provides a constant factor improvement over Bollobás's classical bound.

Abstract

A well-known result of Bollob\'as says that if $\{(A_i, B_i)\}_{i=1}^m$ is a set pair system such that $|A_i| \le a$ and $|B_i| \le b$ for $1 \le i \le m$, and $A_i \cap B_j \ne \emptyset$ if and only if $i \ne j$, then $m \le {a+b \choose a}$. F\"uredi, Gy\'arf\'as and Kir\'aly recently initiated the study of such systems with the additional property that $|A_i \cap B_j| = 1$ for all $i \ne j$. Confirming a conjecture of theirs, we show that this extra condition allows an improvement of the upper bound (at least) by a constant factor.