Hard examples for bounded depth frege

We prove exponential lower bounds on the size of a bounded depth Frege proof of a Tseitin graph-based contradiction, whenever the underlying graph is an expander. This is the first example of a contradiction, naturally formalized as a 3-CNF, that has no short bounded depth Frege proofs. Previously, lower bounds of this type were known only for the pigeonhole principle [18, 17], and for Tseitin contradictions based on complete graphs [19].Our proof is a novel reduction of a Tseitin formula of an expander graph to the pigeonhole principle, in a manner resembling that done by Fu and Urquhart [19] for complete graphs.In the proof we introduce a general method for removing extension variables without significantly increasing the proof size, which may be interesting in its own right.