Notes on switching lemmas

We prove three switching lemmas, for random restrictions for which variables are set independently; for random restrictions where variables are set in blocks (both due to H̊astad [3]); and for a distribution appropriate for the bijective pigeonhole principle [2, 4]. The proofs are based on Beame’s version [1] of Razborov’s proof of the switching lemma in [5], except using families of weighted restrictions rather than families of restrictions which are all the same size. This follows a suggestion of Beame in [1]. The result is something between H̊astad’s and Razborov’s methods of proof. We use probabilistic arguments rather than counting ones, in a similar way to H̊astad, but rather than doing induction on the terms in our formula with an inductive hypothesis involving conditional probability, as H̊astad does, we explicitly build one function to bound the probabilities for the whole formula.