$3$-uniform hypergraphs without a cycle of length five

Abstract

In this paper we show that the maximum number of hyperedges in a 3-uniform hypergraph on n vertices without a (Berge) cycle of length five is less than (0.254 + o(1))n 3/2 , improving an estimate of Bollobás and Győri.We obtain this result by showing that not many 3-paths can start from certain subgraphs of the shadow.

A note on the Turán number of a Berge odd cycle

“…They showed ex 3 (n, Berge-C 5 ) ≤ √ 2n 3/2 + 4.5n. This bound was improved to (0.254 + o(1))n 3/2 by Ergemlidze, Győri and Methuku [5]. For cycles of any length, Győri and Lemons [15,16] proved ex r (n, Berge-C k ) = O(n 1+1/⌊k/2⌋ ).…”