Abstract
The Ramsey number r(K 3,Q n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r(K 3,Q n )=2n+1−1 for every n∈ℕ, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K 3,Q n )⩽7000·2n. Here we show that r(K 3,Q n )=(1+o(1))2n+1 as n→∞.
Similar content being viewed by others
References
P. Allen, G. Brightwell and J. Skokan: Ramsey-goodness — and otherwise, Combinatorica 33 (2013), 125–160.
N. Alon, P. Seymour and R. Thomas: A separator theorem for nonplanar graphs, J. Amer. Math. Soc. 3 (1990), 801–808.
S. Brandt: Expanding graphs and Ramsey numbers, available at Freie Universität, Berlin preprint server, ftp://ftp.math.fu-berlin.de/pub/math/publ/pre/1996/pr-a-96-24.ps (1996).
S. A. Burr: Ramsey numbers involving graphs with long suspended paths, J. London Math. Soc. 24 (1981), 405–413.
S. A. Burr and P. Erdős: Generalizations of a Ramsey-theoretic result of Chvátal, J. Graph Theory 7 (1983), 39–51.
V. Chvátal: Tree-complete graph Ramsey numbers, J. Graph Theory 1 (1977), 93.
D. Conlon: Hypergraph packing and sparse bipartite Ramsey numbers, Combin. Probab. Computing 18 (2009), 913–923.
D. Conlon, J. Fox, C. Lee and B. Sudakov: Ramsey numbers of cubes versus cliques, Combinatorica, to appear.
J. Fox and B. Sudakov: Density theorems for bipartite graphs and related Ramseytype results, Combinatorica 29 (2009), 153–196.
V. Nikiforov and C. C. Rousseau: Ramsey goodness and beyond, Combinatorica 29 (2009), 227–262.
F. P. Ramsey: On a Problem of Formal Logic, Proc. London Math. Soc. 30 (1930), 264–286.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by: CNPq bolsas PDJ (GFP, SG, DS), a CNPq bolsa de Produtividade em Pesquisa (RM). This work was carried out during a visit of JS to IMPA in November 2012
Rights and permissions
About this article
Cite this article
Pontiveros, G.F., Griffiths, S., Morris, R. et al. On the Ramsey number of the triangle and the cube. Combinatorica 36, 71–89 (2016). https://doi.org/10.1007/s00493-015-3089-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-015-3089-8