Degree complexity for a modified pigeonhole principle

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 optimal degree. Finally we prove a simple Lemma giving a simulation of Resolution by Polynomial Calculus.