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We derive some combinatorial formulas related to the diagonal Ramsey numbers R (k). Each formula is a statement of the form "F (n, k) = 0 if and only if n R (k), " where F (n, k) is a combinatorial expression which depends on n and k. Our work is closely related to a recent work by De Loera and Wesley.
Pakawut Jiradilok (Wed,) studied this question.