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Proof Complexity of Paris-Harrington Tautologies

Presented by: 
N Galesi Università degli Studi di Roma La Sapienza
Date: 
Monday 26th March 2012 - 11:00 to 11:30
Venue: 
INI Seminar Room 1
Abstract: 
We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bicolorings of graphs. We prove a conditional lower bound in Resolution and a upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in Res(2). We show that under such assumption, there is no refutation of the Paris-Harrington formulas of size quasi-polynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow-up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced Ramsey principle for triangles is established. This is obtained by adapting some constructions due to Erdo ̋s and Mills.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons