Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey’s Theorem. For Paris-Harrington,  prove a non-trivial conditional lor bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lor bound is conditional on a (very reasonable) hardness assumption for a ak (quasi-polynomial) Pigeonhole principle in R es (2).  show that under such an 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 lor 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 off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdős and Mills.  prove a non-trivial Resolution lor bound for a family of such off-diagonal Ramsey principles.
['Lorenzo Carlucci', 'Nicola Galesi', 'Massimo Lauria']
I am writing a research paper on a idea On the Proof Complexity of Paris-Harrington and Off-Diagonal Ramsey Tautologies
[{"from": "human", "value": "I am writing a research paper on a idea On the Proof Complexity of Paris-Harrington and Off-Diagonal Ramsey Tautologies"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the proof complexity of Paris-Harrington\u2019s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey\u2019s Theorem. For Paris-Harrington,  prove a non-trivial conditional lor bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lor bound is conditional on a (very reasonable) hardness assumption for a ak (quasi-polynomial) Pigeonhole principle in R es (2).  show that under such an 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 lor 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 off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erd\u0151s and Mills.  prove a non-trivial Resolution lor bound for a family of such off-diagonal Ramsey principles."}]