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Unprovability of sharp versions of Friedman’s sineprinciple
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 135
, 2007
"... For every n ≥ 1 and every function F of one argument, we introduce the statement SPn F:“forall m, thereisNsuch that for any set A = {a1,a2,...,aN} of rational numbers, there is H ⊆ A of size m such that for any two nelement subsets ai1 <ai2 < ·· · <ain and ai1 <ak2 < ·· · <akn in ..."
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For every n ≥ 1 and every function F of one argument, we introduce the statement SPn F:“forall m, thereisNsuch that for any set A = {a1,a2,...,aN} of rational numbers, there is H ⊆ A of size m such that for any two nelement subsets ai1 <ai2 < ·· · <ain and ai1 <ak2 < ·· · <akn in H, wehave sin(ai1 · ai2 ···ain) − sin(ai1 · ak2 ···akn)  <F(i1)”. We prove that for n ≥ 2 and any function F(x) eventually dominated by ( 2 3)log(n−1) (x) n+1, the principle SPF is not provable in IΣn. In particular, the statement ∀nSPn ( 2 3)log(n−1) is not provable in Peano Arithmetic. In dimension 2, the result is: IΣ1 does not prove SP2 F,whereF(x)=(2 3) A−1 (x) √ x and A−1 is the inverse of the Ackermann function.
Sharp Thresholds for the Phase Transition between Primitive Recursive and
"... We compute the sharp thresholds on g at which glarge and gregressive Ramsey numbers cease to be primitive recursive and become Ackermannian. We also identify the threshold below which gregressive colorings have usual Ramsey numbers, that is, admit homogeneous, rather than just minhomogeneous set ..."
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We compute the sharp thresholds on g at which glarge and gregressive Ramsey numbers cease to be primitive recursive and become Ackermannian. We also identify the threshold below which gregressive colorings have usual Ramsey numbers, that is, admit homogeneous, rather than just minhomogeneous sets. Key words: ParisHarrington theorem, KanamoriMcAloon theorem, Ackermannian functions, rapidly growing Ramsey numbers
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"... Exact unprovability results for compound wellquasiordered combinatorial classes ..."
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Exact unprovability results for compound wellquasiordered combinatorial classes