On the local density problem for graphs of given odd‐girth

Abstract

Erdős conjectured that every n‐vertex triangle‐free graph contains a subset of ⌊n/2⌋ vertices that spans at most n2/50 edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so‐called Andrásfai graphs. As a consequence, Erdős' conjecture holds for every triangle‐free graph G with minimum degree δ(G)>10n/29 and if χ(G)≤3 the degree condition can be relaxed to δ(G)>n/3. In fact, we obtain a more general result for graphs of higher odd‐girth.

K_4-free graphs have sparse halves

“…Combined with p21t ´4m ´12nq `p9t `4m ´12nq " 30t ´24n ď 6t this implies 21t ´4m ´12n ď 6t and substituting this back into (2.6) we learn p3t ´4mq 2 ď 6tp9t `4m ´12nq , i.e., n ď 3 4 t `1 3 m ´1 72t p3t ´4mq 2 ď 3 4 t `1 3 m . As the term in the middle simplifies to 2 3 m `5 8 t ´2m 2 9t this completes the proof of (2.1). Moreover n ď 3 4 n ´1 3 m implies m ě 3 4 n.…”

“…Throughout the rest of this article we shall call an n-vertex graph G extremal if it is K 4 -free and every set X Ď V pGq of size |X| " t 1 2 nu spans at least The reason why this statement implies Theorem 1.1 is as follows: If n is odd we construct a graph H by replacing every vertex x of G by two new vertices x 1 , x 2 and every edge xy P EpGq by all four possible edges from tx 1 , x 2 u to ty 1 , y 2 u. Evidently, H is still K 4 -free and it is not difficult to verify that any n vertices of H span at least 2 9 n 2 edges. So Proposition 2.1 tells us that 2n is divisible by 3 and that H is a tripartite Turán graph.…”

“…see [7]). This so-called sparse halves conjecture attracted the attention of many researchers [2,11,12,17] but even despite the recent investigations of Razborov [18] the problem remains elusive.…”

Andrásfai and Vega graphs in Ramsey–Turán theory

“…These considerations reveal that balanced blow‐ups of Andrásfai graphs are universal in the class of triangle‐free Cayley graphs whose density is larger than 1/3. Somewhat relatedly, a finite graph is a subgraph of a blow‐up of an Andrásfai graph if and only if it is isomorphic to a subgraph of the infinite triangle‐free Cayley graph , see [4]*Lemma 2.1. We proceed with three well‐known, useful properties of Andrásfai graphs.…”

The Spectrum of Triangle-free Graphs

“…Erdős offered a $250 reward for proving this conjecture. There has been progress on this conjecture in various directions [4,11,13,14,16]. Most recently, Razborov [16] proved that every triangle-free graph on n vertices has an induced subgraph on n/2 vertices with at most (27/1024)n 2 edges.…”