We show that recognizing the $K_3$-freeness and $K_4$-freeness of
graphs is hard, respectively, for two-player nondeterministic
communication protocols with exponentially many partitions and for
nondeterministic (syntactic) read-$s$ times branching programs.

The key ingradient is a generalization of a coloring lemma, due to
Papadimitriou and Sipser, which says that for every ... more >>>