Turán problems and shadows I: Paths and cycles

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Abstract

A k-path is a hypergraph Pk={e1,e2,,ek} such that |eiej|=1 if |ji|=1 and eiej= otherwise. A k-cycle is a hypergraph Ck={e1,e2,,ek} obtained from a (k1)-path {e1,e2,,ek1} by adding an edge ek that shares one vertex with e1, another vertex with ek1 and is disjoint from the other edges.

Let exr(n,G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We prove that for fixed r3, k4 and (k,r)(4,3), for large enough n:exr(n,Pk)=exr(n,Ck)=(nr)(nk12r)+{0if k is odd(nk122r2)if k is even and we characterize all the extremal r-graphs. We also solve the case (k,r)=(4,3), which needs a special treatment. The case k=3 was settled by Frankl and Füredi.

This work is the next step in a long line of research beginning with conjectures of Erdős and Sós from the early 1970s. In particular, we extend the work (and settle a conjecture) of Füredi, Jiang and Seiver who solved this problem for Pk when r4 and of Füredi and Jiang who solved it for Ck when r5. They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.

Keywords

Hypergraph Turán number
Paths
Cycles
Uniform hypergraphs

Cited by (0)

1

Research of this author is supported in part by NSF grants DMS-0965587 and DMS-1266016 and by grant 12-01-00631 of the Russian Foundation for Basic Research.

2

Research partially supported by NSF grants DMS-0969092 and DMS-1300138.

3

Research supported by NSF grants DMS-1362650 and DMS-1101489.