Geodesic
On Tur\'an's $(3,4)$-Problem with Forbidden Subgraphs
Matematičeskie zametki, Tome 95 (2014) no. 2, pp. 271-281.

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We identify three $3$-graphs on five vertices that are missing in all known extremal configurations for Turán's $(3,4)$-problem and prove Turán'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 Turán'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.
Keywords: Turán's $(3,4)$-problem, $3$-graph, hypergraph, forbidden subgraph. 
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A. A. Razborov. On Tur\'an's $(3,4)$-Problem with Forbidden Subgraphs. Matematičeskie zametki, Tome 95 (2014) no. 2, pp. 271-281. http://geodesic.mathdoc.fr/item/MZM_2014_95_2_a9/

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