# Upper bound for Ramsey number for the hypercube

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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 \( r(Q_n) \leq n 2^{2n+5} \).