Upper bound for Ramsey number for the hypercube

For two graphs \( G\) and \( H\), let \( r(G,H)\) denote the smallest integer \( m\) satisfying the property that if the edges of the complete graph \( K_m\) are colored in red and blue, then there is either a subgraph isomorphic to \( G\) with all red edges or a subgraph isomorphic to \( H\) with all blue edges. The classical Ramsey numbers are those for the complete graphs and are denoted by \( r(s,t)= r(K_s, K_t)\). When \(G=H\), we write \(r(G) = r(G,G)\).

A Ramsey problem for \(n\)-cubes (proposed by Burr and Erdös [2])

Let \(Q_n\) denote the \(n\)-cube on \(2^n\) vertices and \(n2^{n-1}\) edges. Prove that \[ r(Q_n) \leq c 2^n .\]

The best known upper bound for \( r(Q_n)\) is due to Fox and Sudakov [1], who showed that \( r(Q_n) \leq n 2^{2n+5} \).

Bibliography
1 J. Fox and B. Sudakov, Density theorems for bipartite graphs and related Ramsey-type results, Combinatorica 29(2) (2009), 153-196.

2 S. A. Burr and P. Erdös, On the magnitude of generalized Ramsey numbers for graphs, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdös on his 60th birthday), Vol. I; Colloq. Math. Soc. János Bolyai, Vol. 10, 215-240, North-Holland, Amsterdam, 1975.