Avoiding long Berge cycles

Let n≥k≥r+3 and H be an n-vertex r-uniform hypergraph. We show that if |H|>[Formula presented](k−1r) then H contains a Berge cycle of length at least k. This bound is tight when k−2 divides n−1. We also show that the bound is attained only for connected r-uniform hypergraphs in which every block is the complete hypergraph Kk−1 (r). © 201

Avoiding long Berge cycles, the missing cases k=r+1 and k=r+2

The maximum size of anr-uniform hypergraph without a Berge cycle of length at leastkhas been determined for allk >= r+ 3 by Furedi, Kostochka and Luo and fork<r(andk=r, asymptotically) by Kostochka and Luo. In this paper we settle the remaining cases:k=r+ 1 andk=r+ 2, proving a conjecture of Furedi, Kostochka and Luo