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Adjacent Ramsey Theory Harvey M. Friedman 10/2/99 This concerns Special FOM e-Mail Postings number 36 and 37. Let k 2 and f:N k [1,k] and n 1 be such that there is no x 1 < . .. < x k+1 £ n such that f(x 1 ,...,x k ) = f(x 1 ,...,x k+1 ). Then we want to find g:N k+1 [1,3] such that there is no x 1 < ... < x k+2 £ n such that g(x 1 ,...,x k+1 ) = g(x 2 ,...,x k+2 ). This reducees adjacent Ramsey in k dimensions with k colors to adjacent Ramsey in k+1 dimensions with 3 colors. Let x 1 < . .. < x k+1 £ n be given. Look at t 1 ,...,t k+1 , where t i = f(x 1 ,...,x i-1 ,x i+1 ,...,x k+1 ). case 1. t 1 ,...,t k+1 all = or strictly monotone. Equality violates the hypotheses, whereas strict monotonicity is impossible because f goes into [1,k]. case 2. first i such that t i ,t i+1 ,t i+2 not strictly monotone is even. Set g(x 1 ,...,x k+1 ) = 1. case 3. first i such that t i ,t i+1 ,t i+2 is odd and t i < t i+1 . Set g(x 1 ,...,x k+1 ) = 2. case 4. first i such that t i ,t i+1 ,t i+1 is odd and t i t i+1 . Set g(x 1 ,...,x k+1 ) = 3. Now let x 1 < . .. < x k+2 £ n be such that g(x 1 ,...,x k+1 ) = g(x 2 ,...,x k+2 ).

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