Short proofs are narrow-resolution made simple

We develop a general strategy for proving width lower bounds, which follows Haken´s original proof technique but is now simple and clear. It reveals that large width is implied by certain natural expansion properties of the clauses (axioms) of the tautology in question. We show that in the classical examples of the Pigeonhole principle, Tseitin graph tautologies, and random k-CNFs, these expansion properties are quite simple to prove. We further illustrate the power of this approach by proving new exponential lower bounds to two different restricted versions of the pigeon-hole principle. One restriction allows the encoding of the principle to use arbitrarily many extension variables in a structured way. The second restriction allows every pigeon to choose a hole from some constant size set of holes