A short note on supersaturation for oddtown and eventown

TL;DR

This paper investigates lower bounds on the number of pairs with odd intersections in collections of subsets, establishing optimal bounds for specific collection sizes and exploring related intersection minimization problems.

Contribution

It provides tight bounds for the number of odd-intersection pairs in collections of even and odd-sized subsets, extending understanding of supersaturation phenomena in combinatorics.

Findings

Proved lower bounds for odd intersection pairs in even-sized collections.

Established minimal number of odd intersection pairs in odd-sized collections.

Connected intersection minimization problems to supersaturation results.

Abstract

Given a collection $\mathcal{A}$ of subsets of an $n$ element set, let $\text{op}(\mathcal{A})$ denote the number of distinct pairs $A,B \in \mathcal{A}$ for which $|A \cap B|$ is odd. For $s \in \{1,2\}$, we prove $\text{op}(\mathcal{A}) \geq s \cdot 2^{\lfloor n/2 \rfloor-1}$ for any collection $\mathcal{A}$ of $2^{\lfloor n/2 \rfloor}+s$ even-sized subsets of an $n$ element set. We also prove $\text{op}(\mathcal{A}) \geq 3$ for any collection $\mathcal{A}$ of $n+1$ odd-sized subsets of an $n$ element set that. Moreover, we show that both of these results are best possible. We then consider larger collections of odd-sized and even-sized sets respectively and explore the connection to minimizing the number of pairwise intersections of size exactly $k-2$ amongst collections of size $k$ subsets from an $n$ element set.