More about sparse halves in triangle-free graphs

TL;DR

This paper advances understanding of Erdos's conjecture on triangle-free graphs by providing new bounds and proving the conjecture for specific graph classes, including girth ≥ 5, large independence number, and strongly regular graphs.

Contribution

It introduces a new bound of 27/1024 n^2 edges and proves the conjecture for several special classes of triangle-free graphs.

Findings

Established a new edge bound of 27/1024 n^2 in general case.

Proved the conjecture for graphs with girth ≥ 5.

Confirmed the conjecture for graphs with independence number ≥ 2n/5 and for strongly regular graphs.

Abstract

One of Erdos's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish the new bound $\frac{27}{1024}n^2$ on the number of edges in general case. We completely prove the conjecture for graphs of girth $\geq 5$, for graphs with independence number $\geq 2n/5$ and for strongly regular graphs. Each of these three classes includes both known (conjectured) extremal configurations, the 5-cycle and the Petersen graph.