New Turán Densities for 3-Graphs

Abstract

If F is a family of graphs then the Turán density of F is determined by the minimum chromatic number of the members of F.The situation for Turán densities of 3-graphs is far more complex and still very unclear. Our aim in this paper is to present new exact Turán densities for individual and finite families of 3-graphs, in many cases we are also able to give corresponding stability results. As well as providing new examples of individual 3-graphs with Turán densities equal to 2/9, 4/9, 5/9 and 3/4 we also give examples of irrational Turán densities for finite families of 3-graphs, disproving a conjecture of Chung and Graham. (Pikhurko has independently disproved this conjecture by a very different method.)A central question in this area, known as Turán's problem, is to determine the Turán density of K (3) 4

An Exact Turán Result for Tripartite 3-Graphs

“…If V (G − {a, b, c}) = {s, t, u, v, w} then we may suppose that stu, stv, uvw, abc ∈ G. Since G is K − 4 -free it does not contain suv or tuv. Moreover it contains at most 3 edges from {u, v, w} (2) ×{a, b, c} and at most 5 edges from {s, t, u, v, w}× {a, b, c} (2) . Since G is F 6 -free it contains no edges from {s, t} × {w} × {a, b, c}.…”

On the algebraic and topological structure of the set of Turán densities

“…This practically represents all that is known about the topological structure of Π (r) ∞ . On the algebraic side, it was proved by Baber and Talbot [2] that Π (3) fin contains irrational numbers, disproving a conjecture of Chung and Graham [10]. Pikhurko independently proved the following more general result.…”

“…Let Π ∞ . The Erdős-Stone-Simonovits theorem ( [17], [16]) completely determines the set Π (2) ∞ . In fact, Π (2)…”

“…The proof uses flag algebras, introduced and developed by Razborov [45]. Flag algebras have been successfully used for computing Turán densities in certain special cases ( [46], [2], [19]), and also for solving several open questions in graph theory ( [27], [24], [28], [35], [3], [22]).…”

Tournaments, 4-uniform hypergraphs, and an exact extremal result

“…[28]). Many further results on extremal numbers for small cliques in hypergraphs can be found, for example, in [6,17,18,20,25,26,30,32,34].…”