On Turan's (3,4)-problem with forbidden configurations

TL;DR

This paper advances the understanding of Turan's (3,4)-problem by identifying specific forbidden configurations and proving the conjecture for 3-graphs excluding these configurations, using a novel indirect interpretation technique.

Contribution

It introduces a new 'indirect interpretation' method to analyze extremal 3-graphs and proves Turan's conjecture under additional forbidden subgraph constraints.

Findings

Identified three key forbidden 3-graphs missing in extremal configurations.

Proved Turan's conjecture for 3-graphs excluding these forbidden configurations.

Used flag algebra calculations to support results on other forbidden subgraphs.

Abstract

We identify three 3-graphs on five vertices each missing in all known extremal configurations for Turan's (3,4)-problem and prove Turan's conjecture for 3-graphs that are additionally known not to contain any induced copies of these 3-graphs. Our argument is based on an (apparently) new technique of "indirect interpretation" that allows us to retrieve additional structure from hypothetical counterexamples to Turan's conjecture, but in rather loose and limited sense. We also include two miscellaneous calculations in flag algebras that prove similar results about some other additional forbidden subgraphs.