PA incomp082910

PA incomp082910 - 1 ADJACENT RAMSEY THEORY by Harvey M....

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1 ADJACENT RAMSEY THEORY by Harvey M. Friedman* Department of Mathematics Ohio State University January 23, 2008 July 27, 2010 August 29, 2010 DRAFT Abstract. We introduce Adjacent Ramsey Theory, which investigates solutions to the shift equation f(x 1 ,...,x k ) = f(x 2 ,.... ,x k+1 ) and the shift inequality f(x 1 ,...,x k ) f(x 2 ,...,x k+1 ), in the context of functions f:N k N r . Existence theorems are proved, and shown to have strong metamathematical properties such as unprovability within Peano Arithmetic. Adjacent Ramsey Theorems represent a new level of naturalness and simplicity for independence results at the level of Peano Arithmetic. 1. f(x 1 ,...,x k ) = f(x 2 ,...,x k+1 ). 2. f(x 1 ,...,x k ) f(x 2 ,...,x k+1 ). 3. f computable. 4. f limited. 5. f finite. 1. f(x 1 ,...,x k ) = f(x 2 ,...,x k+1 ). TO BE COMPLETED. We use N for the set of all nonnegative integers, and [t] for {0,. ..,t-1}, t N. THEOREM 1.1. For all f:N k [2], there exist distinct x 1 ,...,x k+1 such that f(x 1 ,...,x k ) = f(x 2 ,...,x k+1 ). THEOREM 1.2. For all f:N k [2], there exist x 1 < . .. < x k+1 such that f(x 1 ,...,x k ) = f(x 2 ,...,x k+1 ). THEOREM 1.3. For all f:N k [3], there exist distinct x 1 ,...,x k+1 such that f(x 1 ,...,x k ) = f(x 2 ,...,x k+1 ). THEOREM 1.4. For all f:N k [t], there exist x 1 < . .. < x k+1 such that f(x 1 ,...,x k ) = f(x 2 ,...,x k+1 ).
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2 We have proved that theorems 1.2, 1.3 correspond to roughly k fold iterated exponentiation. Theorem 1.1 however has a small upper bound. After we obtained our results, we became aware of Shift graphs and lower bounds on Ramsey numbers rk(l;r), by D. Duffus, H. Lefmann, and V. Rodl, Discrete Mathematics, volume 137, Issues 1-3 (January 1995), 177-187 which has considerable overlap with Theorems 1.1 - 1.4, but stated in very different language. THEOREM 1.5. For all even k 1 and f:[k+1] k [2], there exist distinct x 1 ,...,x k+1 such that f(x 1 ,...,x k ) = f(x 2 ,...,x k+1 ). For all odd k 1 and f:[k+2] k [2], there exist distinct x 1 ,...,x k+1 such that f(x 1 ,...,x k ) = f(x 2 ,...,x k+1 ). Proof: Fix k 1, k even. Consider the sequence (0,. ..,k-1), (1,. ..,k),(2,. ..,k,0),(3,. ..,k,0,1),. ..,(k,0,. ..,k- 1),(0,. ..,k). There are k+2 terms, which is an even number of terms. Hence the values of f at some adjacent pair must be equal. Fix k 1, k odd. Consider the sequence (0,. ..,k-1), (1,. ..,k),(2,. ..,k,k+1),(3,. ..,k,k+1,0),. ..,(k+1,0,. ..,k- 2),(0,. ..,k-1). There are k+3 terms, which is an even number of terms. Hence the values of f at some adjacent pair must be equal. QED THEOREM 1.6. Theorem 1.5 is best possible in the sense that k+1 cannot be replaced by k, and k+2 cannot be replaced by k+1. Proof: The first claim is obvious. For the second claim,
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PA incomp082910 - 1 ADJACENT RAMSEY THEORY by Harvey M....

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