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A Ramsey-type result for the hypercube Rados Radoicic

Summary: A Ramsey-type result for the hypercube
Noga Alon
Rados RadoiciŽc
Benny Sudakov
Jan VondrŽak §
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 k-edge coloring of
a sufficiently large Qn contains a monochromatic copy of H.
1 Introduction
Let Qn denote the graph of the n-dimensional hypercube whose vertex set is {0, 1}n and two vertices
are adjacent if they differ in exactly one coordinate. Ramsey and TurŽan-type 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 n-dimensional
hypercube are necessary to imply the existence of a 4-cycle. Erdos conjectured that (1
2 + o(1))n2n-1
edges are enough to force the appearance of C4. A similar question was posed for the existence of a


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics