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arXiv · 2104.09406

More about sparse halves in triangle-free graphs

Alexander Razborov
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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.

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Alexander Razborov. 2021-07-28. More about sparse halves in triangle-free graphs. https://doi.org/10.1070/sm9615

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