Abstract
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.
Similar content being viewed by others
References
B. Andrásfai: Über ein Extremalproblem der Graphentheorie, Acta Math. Acad. Sci. Hungar. 13 (1962), 443–455.
B. Bollobás and P. Erdős: On a Ramsey-Turán type problem, J. Combinatorial Theory Ser. B 21 (1976), 166–168.
S. Brandt: Triangle-free graphs whose independence number equals the degree, Discrete Math. 310 (2010), 662–669.
S. Brandt and T. Pisanski:. Another infinite sequence of dense triangle-free graphs. Electron. J. Combin. 5 (1998), #R43.
S. Brandt and S. Thomassé: Dense triangle-free graphs are four-colorable: A solution to the Erdős-Simonovits problem, available from Thomassé’s webpage at http://perso.ens-lyon.fr/stephan.thomasse/.
P. Erdős, A. Hajnal, V. T. Sós and E. Szemerédi: More results on Ramsey-Turaan type problems, Combinatorica 3 (1983), 69–81.
P. Erdős and M. Simonovits: A limit theorem in graph theory, Studia Sci. Math. Hungar. 1 (1966), 51–57.
P. Erdős and M. Simonovits: On a valence problem in extremal graph theory, Discrete Math. 5 (1973), 323–334.
P. Erdős and V. T. Sós: Some remarks on Ramsey’s and Turán’s theorem, in: Colloquia mathematica societatis János Bolyai, 4. Combinatorial theory and its applications, Balatonfüred (Hungary), 1969, 395–404, 1970.
P. Erdős and A. H. Stone: On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), 1087–1091.
Ph. Hall: On representatives of subsets, Journal of the London Mathematical Society 10 (1935), 26–30.
T. Łuczak, J. Polcyn and Chr. Reiher: Andrásfai and Vega graphs in Ramsey-Turán theory, Journal of Graph Theory 98 (2021), 57–80.
T. Łuczak: On the structure of triangle-free graphs of large minimum degree, Combinatorica 26 (2006), 489–493.
C. M. Lüders and Chr. Reiher: The Ramsey-Turán problem for cliques, Israel J. Math. 230 (2019), 613–652.
W. Mantel: Problem 28 (solution by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff), Wiskundige Opgaven 10 (1907), 60–61.
M. Simonovits and V. T. Sós: Ramsey-Turán theory, Discrete Math. 229 (2001), 293–340.
E. Szemerédi:. On graphs containing no complete subgraph with 4 vertices. Mat. Lapok 23 (1972), 113–116 (1973).
P. Turán: On an extremal problem in graph theory, Matematikai és Fizikai Lapok (in Hungarian) (1948), 436–452.
A. A. Zykov: On some properties of linear complexes, Mat. Sbornik N.S. 24 (1949), 163–188.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by National Science Centre, Poland, grant 2017/27/B/ST1/00873.
Rights and permissions
About this article
Cite this article
Łuczak, T., Polcyn, J. & Reiher, C. On the Ramsey-Turán Density of Triangles. Combinatorica 42, 115–136 (2022). https://doi.org/10.1007/s00493-021-4340-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-021-4340-0