A standard proof of Schur's Theorem yields that any $r$-coloring $\{1,2,\dots,R_r-1\}$ a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, minimum $N$ such complete graph on vertices triangle. We explore generalizations and modifications this result in higher dimensional integer lattices, showing particular if $k\geq d+1$, then $\{1,2,\dots,R_r(k)^d-1\}^d$...

Recent results established exponential lower bounds for the length of any Resolution proof for the weak pigeonhole principle. More formally, it was proved that any Resolution proof for the weak pigeonhole principle, with n holes and any number of pigeons, is of length fl(2 ), (for a constant e = 1/3). One corollary is that certain propositional formulations of the statement P / NP do not have s...

The pigeonhole principle asserts that there is no injective mapping from m pigeons to n holes as long as m > n. It is amazingly simple, expresses one of the most basic primitives in mathematics and Theoretical Computer Science (counting) and, for these reasons, is probably the most extensively studied combinatorial principle. In this survey we try to summarize what is known about its proof comp...

The Pigeonhole Principle (PHP) has been one of the most appealing methods of solving combinatorial optimization problems. Variations of the Pigeonhole Principle, sometimes called the “Hidden” Pigeonhole Principle (HPHP), are even more powerful and often produce the most elegant solutions to nontrivial problems. However, some Operations Research approaches, such as the Linear Programming Relaxat...

We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give lower bounds for tree-like resolution proofs. We present a normal form for resolution proofs of pigeonhole principles based on a new monotone resolution rule.