Claims

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2003_04_23_rodl: Rödl's 2003 paper proves that the least weight of a monochromatic clique forced by every two-coloring of the pairs of {2, ..., n} tends to infinity with n, so the answer is yes; the paper is not held here.

2011_12_07_conlon_fox_sudakov: Theorem 1.1 of Conlon, Fox and Sudakov (Duke Math. J. 2013): for large n, every two-coloring of the pairs of {2, ..., n} has a monochromatic clique of weight at least 2^(-8) log log log n, so the answer is yes.

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