The fine intersection problem for Steiner triple systems

TL;DR

This paper investigates the fine intersection problem for Steiner triple systems, establishing that for certain orders, the set of possible intersection sizes grows at least on the order of v^3, improving previous bounds.

Contribution

The paper proves that for v ≡ 1 or 3 mod 6, the set of possible intersection pairs has size at least proportional to v^3, advancing understanding of intersection structures.

Findings

|I(v)| = Omega(v^3) for v ≡ 1 or 3 mod 6

Previous bounds only showed |I(v)| = Omega(v^2)

Provides new lower bounds for intersection sizes in Steiner triple systems

Abstract

The intersection of two Steiner triple systems (X,A) and (X,B) is the set A intersect B. The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m,n) such that there exist two Steiner triple systems of order v whose intersection has n blocks over m points. We show that for v = 1 or 3 (mod 6), |I(v)| = Omega(v^3), where previous results only imply that |I(v)| = Omega(v^2).