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On the Ramsey-Turán Density of Triangles


One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graph on n vertices has at most ⌊n2/4⌋ edges. About half a century later Andrásfai studied dense triangle-free graphs and proved that the largest triangle-free graphs on n vertices without independent sets of size αn, where 2/5 ≤ α < 1/2, are blow-ups of the pentagon. More than 50 further years have elapsed since Andrásfai’s work. In this article we make the next step towards understanding the structure of dense triangle-free graphs without large independent sets.

Notably, we determine the maximum size of triangle-free graphs G on n vertices with α(G) ≥ 3n/8 and state a conjecture on the structure of the densest triangle-free graphs G with α(G) > n/3. We remark that the case α(G) α n/3 behaves differently, but due to the work of Brandt this situation is fairly well understood.

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Correspondence to Christian Reiher.

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The first author was partially supported by National Science Centre, Poland, grant 2017/27/B/ST1/00873.

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Łuczak, T., Polcyn, J. & Reiher, C. On the Ramsey-Turán Density of Triangles. Combinatorica 42, 115–136 (2022).

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Mathematics Subject Classification (2010)

  • 05C35
  • 05C69