### Some Ramsey results for the n-cubeSome Ramsey results for the n-cube

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 Author Graham, Ron ♦ Solymosi, Jozsef Source arXiv.org Content type Text File Format PDF Date of Submission 2008-07-09 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Combinatorics ♦ 05D10 ♦ math Abstract In this note we establish a Ramsey-type result for certain subsets of the $n$-dimensional cube. This can then be applied to obtain reasonable bounds on various related structures, such as (partial) Hales-Jewett lines for alphabets of sized 3 and 4, Hilbert cubes in sets of real numbers with small sumsets, "corner" in the integer lattice in the plane, and 3-term geometric progressions in integers. Educational Use Research Learning Resource Type Article Page Count 10