The number $4/9$ is a non-jump for $3$-graphs

TL;DR

This paper proves that the fraction 4/9 is a non-jump value for 3-uniform hypergraphs, introducing a novel construction that challenges previous barriers and supports a conjecture about the smallest non-jump.

Contribution

The authors construct a new hypergraph pattern perturbation that surpasses previous barriers, advancing understanding of non-jumps in 3-graphs.

Findings

4/9 is a non-jump for 3-graphs.

The construction involves high-cogirth Steiner triple systems.

Supports the conjecture that 4/9 is the smallest non-jump.

Abstract

We prove that $4/9$ is a non-jump for $3$-uniform hypergraphs. Our construction perturbs the $ABB$ pattern by inserting, inside the $B$-part, the union of a high-cogirth pair of Steiner triple systems. This goes below the barrier for non-jumps obtainable by Shaw's finite-pattern formulation of the Frankl--R\"odl method introduced in 1984. All results employing this approach use patterns where one of the parts has complete shadow. As the $ABB$ pattern is the smallest one with this property, the value $4/9$ is the natural barrier using this technique, and we conjecture that $4/9$ is the smallest non-jump for $3$-graphs. If our conjecture is true, this would answer (in a very strong form) an old question of Erd\Hos.