Lower bounds to the size of constant-depth propositional proofs
Journal of Symbolic Logic
Exponential lower bounds for the pigeonhole principle
Computational Complexity
Bounded arithmetic, propositional logic, and complexity theory
Bounded arithmetic, propositional logic, and complexity theory
An exponential lower bound to the size of bounded depth Frege proofs of the Pigeonhole Principle
Random Structures & Algorithms
The relative complexity of NP search problems
Journal of Computer and System Sciences
Ramsey's Theorem in Bounded Arithmetic
CSL '90 Proceedings of the 4th Workshop on Computer Science Logic
A new proof of the weak Pigeonhole principle
Journal of Computer and System Sciences - Special issue on STOC 2000
Examples of hard tautologies in the propositional calculus
STOC '81 Proceedings of the thirteenth annual ACM symposium on Theory of computing
The complexity of the pigeonhole principle
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
A lower bound on the size of resolution proofs of the Ramsey theorem
Information Processing Letters
A rank lower bound for cutting planes proofs of ramsey's theorem
SAT'13 Proceedings of the 16th international conference on Theory and Applications of Satisfiability Testing
The complexity of proving that a graph is ramsey
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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A Ramsey statement denoted $${n \longrightarrow (k)^2_2}$$ says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(n k ) and with terms of size $${\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}$$ . Let r k be the minimal n for which the statement holds. We prove that RAM(r k , k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák's work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4 k , k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R.