On the Fon-der-Flaass Interpretation of Extremal Examples for Turán’s (3,4)-problem
نویسنده
چکیده
In 1941, Turán conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turán (3, 4)-graph) is≥ 4/9(1− o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many other examples of this density. Fon-der-Flaass (1988) presented a general construction that converts an arbitrary ~ C4-free orgraph Γ into a Turán (3, 4)-graph. He observed that all Turán-Brown-Kostochka examples result from his construction, and proved the bound ≥ 3/7(1− o(1)) on the edge density of any Turán (3, 4)-graph obtainable in this way. In this paper we establish the optimal bound 4/9(1− o(1)) on the edge density of any Turán (3, 4)-graph resulting from the Fon-derFlaass construction under any of the following assumptions on the undirected graph G underlying the orgraph Γ: • G is complete multipartite; • the edge density of G is ≥ (2/3− 2) for some absolute constant 2 > 0. We are also able to improve Fon-der-Flaass’s bound to 7/16(1− o(1)) without any extra assumptions on Γ. ∗University of Chicago, [email protected]. Part of this work was done while the author was with Steklov Mathematical Institute, supported by the Russian Foundation for Basic Research, and with Toyota Technological Institute at Chicago.
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تاریخ انتشار 2010