We review and comment on a number of results in Ramsey theory obtained recently by the author in collaboration with V. Kanellopoulos, N. Karagiannis and K. Tyros. Among them are density versions of the classical pigeonhole principles of Halpern–Läuchli and Carlson–Simpson. We shall comment on recent progress concerning one fundamental problem in Ramsey theory. It originates from an insightful c...

The purpose of this work is to study, from the viewpoint of first order arithmetic (we have no set variables, the only sets are the arithmetical sets), Ramsey’s Theorem for pairs for a recursive assignment of k-many colors in order to find some principle of classical logic equivalent to it in HA. We proved that it is equivalent in HA to Σ3-LLPO, that is the Limited Lesser Principle of Omniscien...

It is known that the functional and onto variants of the propositional pigeonhole principle have polynomial-size proofs in the weakest deep inference systems. The unrestricted variant has quasipolynomial-size proofs when dag-like behaviour is permitted. Here we match that upper bound in systems free of dagness, and indeed other compression mechanisms, utilising a similar strategy to that of Ats...

We show, under the assumption that factoring is hard, that a model of PV exists in which the polynomial hierarchy does not collapse to the linear hierarchy; that a model of S2 exists in which NP is not in the second level of the linear hierarchy; and that a model of S2 exists in which the polynomial hierarchy collapses to the linear hierarchy and in which the strict version of PH does not colla...

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We define typical forcings encompassing many informal forcing arguments in bounded arithmetic and give general conditions for such to produce models of the universal variant relativized T21. apply this result study relative complexity total (type 2) NP search problems associated finitary combinatorial principles. Complexity theory compares with respect polynomial time many-one or Turing reducti...

Groote and Zantema proved that a particular OBDD computation of the pigeonhole formula has exponential size, and that limited OBDD derivations cannot simulate resolution polynomially. Here we show that an arbitrary OBDD refutation of the pigeonhole formula has exponential size: we prove that for any order of computation at least one intermediate OBDD in the proof has size Ω(1.14n). We also pres...

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