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Some Exact Results and New Asymptotics for Hypergraph Turán Numbers

Published online by Cambridge University Press:  10 July 2002

DHRUV MUBAYI
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA (e-mail: [email protected])

Abstract

Given a family [Fscr ] of r-graphs, let ex(n, [Fscr ]) be the maximum number of edges in an n-vertex r-graph containing no member of [Fscr ]. Let C(r)4 denote the family of r-graphs with distinct edges A, B, C, D, such that AB = CD = Ø and AB = CD. For s1 [les ] … [les ] sr, let K(r) (s1,…,sr) be the complete r-partite r-graph with parts of sizes s1,…,sr.

Füredi conjectured over 15 years ago that ex(n,C(3)4) [les ] (n2) for sufficiently large n. We prove the weaker result

ex(n, {C(3)4, K(3)(1,2,4)}) [les ] (n2).

Generalizing a well-known conjecture for the Turán number of bipartite graphs, we conjecture that

ex(n, K(r)(s1,…,sr)) = Θ(nr−1/s),

where s = Πr−1i=1si. We prove this conjecture when s1 = … = sr−2 = 1 and

(i) sr−1 = 2, (ii) sr−1 = sr = 3, (iii)sr > (sr−1−1)!.

In cases (i) and (ii), we determine the asymptotic value of ex(n,K(r)(s1,…,sr)).

We also provide an explicit construction giving

ex(n,K(3)(2,2,3)) > (1/6−o(1))n8/3.

This improves upon the previous best lower bound of Ω(n29/11) obtained by probabilistic methods. Several related open problems are also presented.

Type
Research Article
Copyright
2002 Cambridge University Press

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