A Ramsey statement denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \longrightarrow (k)^2_2}$$\end{document} 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(nk) and with terms of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}$$\end{document}. Let rk be the minimal n for which the statement holds. We prove that (...) RAM(rk, 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(4k, 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. (shrink)

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res, as well as their tree-like versions, Res⁎. The contradictions we use are natural combinatorial principles: the Least number principle, LNPn and an ordered variant thereof, the Induction principle, IPn.LNPn is known to be easy for Resolution. We prove that its relativization is hard for Resolution, and more generally, the relativization of LNPn iterated d times provides a separation between Res and Res. We prove the (...) same result for the iterated relativization of IPn, where the tree-like variant Res⁎ is considered instead of Res.We go on to provide separations between the parameterized versions of Res and Res. Here we are able again to use the relativization of the LNPn, but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res⁎ and Res⁎. Here, the relativization of IPn will not work as it is, and so we make a vectorizing amendment to it in order to address this shortcoming. (shrink)