### October 25-30 2009

University Residential Center

Bertinoro (Forlì-Cesena), Italy

The workshop wants to offer an opportunity for the communities working in proof theory of arithmetic, in reverse mathematics, in finite and infinite combinatorics of Ramsey theory, in proof complexity and in bounded arithmetic, to be exposed to one another's recent results, methods, and goals.
The limits of computability and of provability are - from different perspectives - the major focus of research in proof theory, reverse mathematics, propositional proof complexity, and bounded arithmetic. Classical combinatorics - and especially Ramsey Theory - is a prominent source of principles and tools for all these areas.

The study of Ramsey-like principles is central in proof theory, both in the study of unprovability as in the reverse mathematics project; in proof complexity some of the most challenging open problems concern the characterization of the hardness of Ramsey-type statements. Constructive bounds for Ramsey theorems are of interest in the study of bounded arithmetic. Ramsey theory is also at the core of some recent successful interactions between logic and combinatorics (logical analysis of ergodic proofs of Ramsey-like statements, provability phase transitions, etc.). Finally, infinite Ramsey theory has found striking applications in topology and in functional analysis (in particular in Banach Space theory).

The goals of the workshop are to stimulate the interaction between researchers in the above-mentioned areas, to enhance the transfer of methods from one area to the other, as well as to set the ground for a unifying view on the logico-combinatorial study of combinatorial principles, such as Ramsey-type statements.