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

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.