Richard P. Stanley Seminar in Combinatorics: Turán numbers of r-graphs on r+1 vertices

SEMINARS, HARVARD-MIT COMBINATORICS

Speaker:

Alexander Sidorenko

Let H_k^r be an r-uniform hypergraph with r+1 vertices and k edges where 3 ≤ k ≤ r+1. It is easy to see that such a hypergraph is unique up to isomorphism. The well-known upper bound on its Turán density is 𝝿(H_k^r) ≤ (k-2)/r. In the case k=3, Frankl and Füredi (1984) used a geometric construction to prove 𝝿(H_3^r) ≥ 2^{1-r}. We use classical results from order statistics going back to Rényi (1953) and a geometric construction to prove 𝝿(H_k^r) ≥ r^{-(1 + 1/(k-2))}.

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