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**1 - 8**of**8**### Phase transitions for monotone increasing sequences, the Erdös-Szekeres theorem and the

"... Motivated by the classical Ramsey for pairs problem in reverse math-ematics we investigate the recursion-theoretic complexity of certain as-sertions which are related to the Erd̈’os-Szekeres-theorem. We show that resulting density principles give rise to Ackermannian growth. We then parameterize the ..."

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Motivated by the classical Ramsey for pairs problem in reverse math-ematics we investigate the recursion-theoretic complexity of certain as-sertions which are related to the Erd̈’os-Szekeres-theorem. We show that resulting density principles give rise to Ackermannian growth. We then parameterize these assertions with respect to a number-theoretic func-tion f and investigate for which functions f Ackermannian growth is still preserved. We show that this is the case for f(i) = sqrt[d]i but not for f(i) = log(i). 1

### Classifying the phase transition threshold for unordered regressive Ramsey numbers

"... Following ideas of Richer (2000) we introduce the notion of unordered regressive Ramsey numbers or unordered Kanamori-McAloon numbers. We show that these are of Ackermannian growth rate. For a given number-theoretic function f we consider unordered f-regressive Ramsey numbers and classify exactly th ..."

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Following ideas of Richer (2000) we introduce the notion of unordered regressive Ramsey numbers or unordered Kanamori-McAloon numbers. We show that these are of Ackermannian growth rate. For a given number-theoretic function f we consider unordered f-regressive Ramsey numbers and classify exactly the threshold for f which gives rise to the Acker-mannian growth rate of the induced f-regressive Ramsey numbers. This threshold coincides with the corresponding threshold for the standard re-gressive Ramsey numbers. Our proof is based on an extension of an argumtent from a corresponding proof in a paper by Kojman,Lee,Omri and Weiermann 2007. 1

### Binary trees and (maximal) order types

"... Abstract. Concerning the set of rooted binary trees, one shows that Higman’s Lemma and Dershowitz’s recursive path ordering can be used for the decision of its maximal order type according to the homeomorphic embedding relation as well as of the order type according to its canonical linearization, w ..."

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Abstract. Concerning the set of rooted binary trees, one shows that Higman’s Lemma and Dershowitz’s recursive path ordering can be used for the decision of its maximal order type according to the homeomorphic embedding relation as well as of the order type according to its canonical linearization, well-known in proof theory as the Feferman-Schütte notation system without terms for addition. This will be done by showing that the ordinal ωn+1 can be found as the (maximal) order type of a set in a cumulative hierarchy of sets of rooted binary trees. 1

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"... Exact unprovability results for compound well-quasi-ordered combinatorial classes ..."

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Exact unprovability results for compound well-quasi-ordered combinatorial classes

### Phase transitions in Proof Theory

"... Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical well-partial-ordering assertions in several familiar theories occurring in the reverse mathematics program. ..."

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Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical well-partial-ordering assertions in several familiar theories occurring in the reverse mathematics program.

### Turchin’s Relation and Subsequence Relation in Loop Approximation

"... The paper studies the subsequence relation through a notion of an intransitive binary relation on words in traces generated by prefix-rewriting systems. The relation was in-troduced in 1988 by V.F. Turchin for loop approximation in supercompilation. We study properties of this relation and introduce ..."

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The paper studies the subsequence relation through a notion of an intransitive binary relation on words in traces generated by prefix-rewriting systems. The relation was in-troduced in 1988 by V.F. Turchin for loop approximation in supercompilation. We study properties of this relation and introduce some refinements of the subsequence relation that inherit the useful features of Turchin’s relation. 1