Triangles in $C_5$-free graphs and Hypergraphs of Girth Six

Abstract

We introduce a new approach and prove that the maximum number of triangles in a C 5 -free graph on n vertices is at mostWe also show a connection to r-uniform hypergraphs without (Berge) cycles of length less than six, and estimate their maximum possible size.

Multicolor Turán numbers

“…Here, the image of an edge becomes a cherry (K 1,2 ). Note that since their short and neat proof on the upper bound, some improvements have been established [10]. Note that these are exactly those graphs where the lower bound of Theorem 3.1 is asymptotically sharp.…”

Extremal problems of double stars

“…For graphs G and H, let n(H, G) denote the number of subgraphs of G isomorphic to H (referred to as copies of H). The first result of this type is due to Zykov [10] (and also independently by Erdős [3]), who determined ex(n, K s , K t ) exactly for all s and t. Then Erdős raised the longstanding conjecture ex(n, C 5 , C 3 ) = ( n 5 ) 5 (where the lower bound is obtained from the uniformly blown up C 5 ). This conjecture was finally verified quarter of a century later by Hatami, Hladký, Král, Norine and Razborov [8] and independently by Grzesik [6].…”

“…On the other hand, the extremal function ex(n, C 3 , C 5 ) was considered by Bollobás and Győri [2]. Their results were subsequently improved in the papers [1], [4] and [5], but the problem of determining the correct asymptotics is still open. The problem of maximizing P ℓ copies in a P k -free graph was investigated in [7].…”

A note on the Turán number of a Berge odd cycle

“…The latter had the stronger bound ex(n, K 3 , C 2k+1 ) ≤ (16k−8) 3 ex(⌈n/2⌉, C 2k ). In case k = 2, the current best bound ex(n, K 3 , C 5 ) ≤ 0.231975n 3/2 is due to Ergemlidze and Methuku [6].…”