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A few new oddtown and eventown problems

Griffin Johnston, Jason O'Neill

TL;DR

This work studies alpha-intersection patterns of families of subsets modulo 2 and modulo 3, systematizing the maximal sizes f_α(n) for α in small dimensional binary vectors. It proves precise and asymptotic results for k=3 and k=4, yielding exact values for most α in F2^3 and asymptotic sqrt{2n} growth for several 4-wise patterns, while also establishing tight duality relations. The authors develop and employ key tools—Trace Lemma, Dual Lemma, and Partition Sum Lemma—to relate patterns, transfer bounds, and construct extremal families, including linear-size constructions and sqrt-growth families. They also explore modulo 3 variants, showing varied behavior including linear, quadratic, and exponential growth regimes, and discuss implications, extensions, and open problems, such as r-uniform variants and general modulo-ℓ settings, with connections to Deza-Frankl-Singhi-type bounds. Overall, the paper advances understanding of higher-order intersection restrictions and their asymptotic landscape, providing a framework for future explorations in extremal set theory and combinatorial design.

Abstract

Given a vector $α= (α_1, \ldots, α_k) \in \mathbb{F}_2^k$, we say a collection of subsets $\mathcal{F}$ satisfies $α$-intersection pattern modulo $2$ if all $i$-wise intersections consisting of $i$ distinct sets from $\mathcal{F}$ have size $α_i \pmod{2}$. In this language, the classical oddtown and eventown problems correspond to vectors $α=(1,0)$ and $α=(0,0)$ respectively. In this paper, we determine the largest such set families of subsets on a $n$-element set with $α$-intersection pattern modulo $2$ for all $α\in \mathbb{F}_2^3$ and all $α\in \mathbb{F}_2^4$ asymptotically. Lastly, we consider the corresponding problem with restrictions modulo $3$.

A few new oddtown and eventown problems

TL;DR

This work studies alpha-intersection patterns of families of subsets modulo 2 and modulo 3, systematizing the maximal sizes f_α(n) for α in small dimensional binary vectors. It proves precise and asymptotic results for k=3 and k=4, yielding exact values for most α in F2^3 and asymptotic sqrt{2n} growth for several 4-wise patterns, while also establishing tight duality relations. The authors develop and employ key tools—Trace Lemma, Dual Lemma, and Partition Sum Lemma—to relate patterns, transfer bounds, and construct extremal families, including linear-size constructions and sqrt-growth families. They also explore modulo 3 variants, showing varied behavior including linear, quadratic, and exponential growth regimes, and discuss implications, extensions, and open problems, such as r-uniform variants and general modulo-ℓ settings, with connections to Deza-Frankl-Singhi-type bounds. Overall, the paper advances understanding of higher-order intersection restrictions and their asymptotic landscape, providing a framework for future explorations in extremal set theory and combinatorial design.

Abstract

Given a vector , we say a collection of subsets satisfies -intersection pattern modulo if all -wise intersections consisting of distinct sets from have size . In this language, the classical oddtown and eventown problems correspond to vectors and respectively. In this paper, we determine the largest such set families of subsets on a -element set with -intersection pattern modulo for all and all asymptotically. Lastly, we consider the corresponding problem with restrictions modulo .
Paper Structure (29 sections, 10 theorems, 35 equations, 4 figures, 3 tables)

This paper contains 29 sections, 10 theorems, 35 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. -wise modulo Patterns
  4. The Problem
  5. The Problem
  6. The Problem
  7. The Problem
  8. The Problem
  9. The Problem
  10. The Problem
  11. The Problem
  12. -wise modulo Patterns
  13. The Problem
  14. The Problem
  15. The Problem
  16. ...and 14 more sections

Key Result

Theorem 1

Let $n\geq 7$. Then

Figures (4)

  • Figure 1: The sets in \ref{['construction:110_family']}
  • Figure 2: The sets in \ref{['construction:001_family']}
  • Figure 3: The sets in \ref{['0100construction']} for $n=7$
  • Figure 4: The sets in \ref{['0001construction']} for $n = 7$

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1: Trace Lemma
  • Lemma 2: Dual Lemma
  • Lemma 3: Partition Sum Lemma
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Theorem A.1: Folklore