On supersaturation for oddtown and eventown

TL;DR

This paper investigates supersaturation problems in oddtown and eventown set systems, providing tight bounds, disproving a conjecture, and extending partial results with new lower bounds and asymptotic analysis.

Contribution

It establishes tight bounds for oddtown supersaturation, disproves O'Neill's conjecture, and offers partial and weaker bounds for eventown, advancing understanding of these combinatorial structures.

Findings

Proved $op(\\mathcal A) \geq s+2$ for oddtown, tight for $s\le n-4$.

Partially proved a conjecture for eventown with bounds depending on $s$ and $n$.

Provided asymptotic lower bounds for large $s$ in both problems.

Abstract

We study the supersaturation problems of oddtown and eventown. Given a family $\mathcal A$ of subsets of an $n$ element set, let $op(\mathcal A)$ denote the number of distinct pairs $A,B\in \mathcal A$ for which $|A \cap B|$ is odd. We show that if $\mathcal A$ consists of $n+s$ odd-sized subsets, then $op(\mathcal A)\geq s+2$, which is tight when $s\le n-4$. This disproves a conjecture by O'Neill on the supersaturation problem of oddtown. For the supersaturation problem of eventown, we show that for large enough $n$, if $\mathcal A$ consists of $2^{\lfloor \frac n 2\rfloor}+s$ even-sized subsets, then $op(\mathcal A)\ge s\cdot2^{\lfloor \frac n 2\rfloor-1} $ for any positive integer $s\le \frac{2^{\lfloor\frac n 8\rfloor}} n$. This partially proves a conjecture by O'Neill on the supersaturation problem of eventown. Previously, the correctness of this conjecture was only verified for $s=1$ and $2$. We further provide a twice weaker lower bound in this conjecture for eventown, that is $op(\mathcal{A})\ge s\cdot 2^{\lfloor n/2\rfloor-2}$ for general $n$ and $s$ by using discrete Fourier analysis. Finally, some asymptotic results for the lower bounds of $op(\mathcal A)$ are given when $s$ is large for both problems.