New upper bounds for the size of set systems with restricted intersections modulo prime powers

TL;DR

This work addresses upper bounds for the size of set systems on $[n]$ that are $L$-avoiding mod $q$ and $L$-intersecting mod $q$, with $q$ a prime power. It advances the theory by combining the linear algebra bound method with separating polynomials and number-theoretic congruences to obtain polynomial upper bounds, notably $m(n,s,q) \le \sum_{j=0}^{q-1} {n\choose j}$, and a refined $(k-1)$-factor bound for $k$-wise intersections under certain residue constraints. Central to the approach are the construction of univariate integer-valued separating polynomials $F_L$ and a Blokhuis-type augmentation using additional polynomials $h_I$, enabling a finite-dimensional bound via linear independence arguments. These results extend prior bounds from primes to prime powers and illuminate the role of modular arithmetic in extremal set theory, with potential implications for broader modular-intersection problems.

Abstract

Let $q=p^α$ be a fixed prime power, $k\geq 2$ be an integer. We give a new upper bound for the size of $k$-wise $q$-modular $L$-avoiding $L$-intersecting set systems, where $L$ is any proper subset of $\{0, \ldots , q-1\}$. Our proof is based on the linear algebra bound method and basic number theory.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Proposition 2.1
• Lemma 2.2