 
Summary: A Ramseytype result for the hypercube
Noga Alon
Rados RadoiciŽc
Benny Sudakov
Jan VondrŽak §
Abstract
We prove that for every fixed k and 5 and for sufficiently large n, every edge coloring of
the hypercube Qn with k colors contains a monochromatic cycle of length 2 . This answers an
open question of Chung. Our techniques provide also a characterization of all subgraphs H of
the hypercube which are Ramsey, i.e., have the property that for every k, any kedge coloring of
a sufficiently large Qn contains a monochromatic copy of H.
1 Introduction
Let Qn denote the graph of the ndimensional hypercube whose vertex set is {0, 1}n and two vertices
are adjacent if they differ in exactly one coordinate. Ramsey and TurŽantype questions concerning
the hypercube were mentioned in a 1984 paper by Erdos [8], but in fact had been considered even
earlier, as in this paper he outlined a collection of "old unsolved problems which had been perhaps
undeservedly neglected". In one of these problems he asked how many edges of an ndimensional
hypercube are necessary to imply the existence of a 4cycle. Erdos conjectured that (1
2 + o(1))n2n1
edges are enough to force the appearance of C4. A similar question was posed for the existence of a
