Cancellative hypergraphs and Steiner triple systems

TL;DR

This paper characterizes the structure of large cancellative triple systems, showing they are close to Steiner triple systems and revealing complex local maxima in their feasible region.

Contribution

It establishes a stability theorem for cancellative triple systems and demonstrates the intricate structure of their feasible region boundary.

Findings

Cancellative triple systems are structurally close to Steiner triple systems under certain conditions.

The boundary of the feasible region has infinitely many local maxima.

First example showing complex local maxima phenomena in such combinatorial systems.

Abstract

A triple system is cancellative if it does not contain three distinct sets $A,B,C$ such that the symmetric difference of $A$ and $B$ is contained in $C$. We show that every cancellative triple system $\mathcal{H}$ that satisfies certain inequality between the sizes of $\mathcal{H}$ and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon.