On the maximal size of $(a,b)$-town$\pmod k$ families

TL;DR

The paper addresses maximal sizes of $(a,b)$-town mod $p$ families, a natural generalization of Oddtown. It introduces $\alpha$-characteristic vectors and bilinear form techniques to bound these families, deriving a universal bound for $a=b$ and a linear bound $|\mathcal{F}|\le n$ for $a\ne b$, with further refinements yielding $|\mathcal{F}|\le n-1$ in infinitely many parameter regimes. A key result is a new bound for $k=3$ when $b-a \equiv 1 \pmod{3}$ and $n \equiv 2 \pmod{3}$, shown to be tight for infinitely many $n$, plus complementary bounds via set complementation. The findings improve upon the general Ray-Chaudhuri–Wilson bounds in several cases and demonstrate that, for fixed $k$, the $|\mathcal{F}|$ bound becomes universes of linear size in many parameter families, with explicit constructions matching some bounds.

Abstract

A family $\mathcal{F}\subseteq\mathcal{P}(n)$ is an $(a,b)$-town$\pmod k$ if all sets in it have cardinality $a\pmod k$ and all pairwise intersections in it have cardinality $b\pmod k$. For $k=2$ the maximal size of such a family is known for each $a,b$, while for $k=3$ only $b-a\equiv 2 \pmod 3$ is fully understood. We provide a bound for $k=3$ when $b-a\equiv 1 \pmod 3$ and $n\equiv 2 \pmod 3$, which turns out to be tight for infinitely many such $n$. We also give sufficient conditions on the parameters $a,b,k,n$, which result in a better bound than the one from general settings by Ray-Chaudhuri--Wilson, in particular showing that this bound occurs infinitely often in a sense where all of $a,b,n$ can vary for a fixed $k$.

Figures (1)

• Figure :

Theorems & Definitions (20)

• Definition 1.1
• Theorem 1.2
• Proposition 1.3
• Proposition 1.4
• Theorem 1.5
• Definition 1.6
• proof : Proof of Proposition \ref{['prop:type-1-2-n-2-bound-intro']}
• Lemma 2.1
• proof
• Proposition 2.2