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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.