As a story, this means that at a party with n persons, there exist two persons who know the same number of people at the party. Proof. For any graph on n vertices, the degrees are integers between 0 and n − 1. Therefore, the only way all degrees could be different is that there is exactly one vertex of each possible degree. In particular, there is a vertex v of degree 0 (with no neighbors) and ...

We consider a modiication of the pigeonhole principle, MPHP, introduced by Goerdt in 7]. Using a technique of Razborov 9] and simpliied by Impagliazzo, Pudll ak and Sgall 8], we prove that any Polynomial Calculus refutation of a set of polynomials encoding the MPHP, requires degree (log n). We also prove that the this lower bound is tight, giving Polynomial Calculus refutations of MPHP of optim...

We show that short bounded-depth Frege proofs of matrix identities, such as PQ = I ⊃ QP = I (over the field of two elements), imply short bounded-depth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential-size bounded-depth Frege proofs, it follows that the propositional version of the matrix principle also requires bounded-depth Frege proofs of e...

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...

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the “finitary” infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alterna...

Since Jeff Paris introduced them in the late seventies [Par78], densities turned out to be useful for studying independence results. Motivated by their simplicity and surprising strength we investigate the combinatorial complexity of two such densities which are strongly related to the pigeonhole principle. The aim is to miniaturise Ramsey’s Theorem for 1-tuples. The first principle uses an unl...

We study the long-standing open problem of giving ∀Σ1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the ∀Σ1 Herbrandized ordering principle is unprovable in a fragment of bounded arithmet...

The pigeonhole principle: "If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole," is an obvious yet fundamental principle of nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. ...

Recently, Raz Raz01] established exponential lower bounds on the size of resolution proofs of the weak pigeonhole principle. We give another proof of this result which leads to better numerical bounds.