Exact solution of the hypergraph Turán problem for k-uniform linear paths

Abstract

A k-uniform linear path of length ℓ, denoted by ℙ (k) , is a family of k-sets {F 1,...,F such that |F i F i+1|=1 for each i and F i F bj = \(\not 0\) whenever |ij|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex k (n, H), is the maximum number of edges in a k-uniform hypergraph \(\mathcal{F}\) on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex k (n, P (k) exactly for all fixed ℓ ≥1, k≥4, and sufficiently large n. We show that

$ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$

.

The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that

$ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$

, and describe the unique extremal family. Stability results on these bounds and some related results are also established.

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Correspondence to Zoltán Füredi.

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Research supported in part by the Hungarian National Science Foundation OTKA, by the National Science Foundation under grant NFS DMS 09-01276, and by a European Research Council Advanced Investigators Grant 267195.

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Füredi, Z., Jiang, T. & Seiver, R. Exact solution of the hypergraph Turán problem for k-uniform linear paths. Combinatorica 34, 299–322 (2014). https://doi.org/10.1007/s00493-014-2838-4

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Mathematics Subject Classification (2010)

  • 05D05
  • 05C65
  • 05C35