Sharpness of the bijumbledness threshold for K_k-factors

Determine whether the order of the bijumbledness parameter p^{k-1}n is sharp for the existence of K_k-factors in pseudorandom graphs when k\geq 4.

Background

The paper surveys known results on K_k-factors in bijumbled graphs. For triangle-factors, triangle-free pseudorandom constructions establish that the order p^2n is sharp, while existing results for general fixed k\geq 3 guarantee K_k-factors under a condition of the form \beta\leq c p^{k-1}n. The authors note that, for k\geq 4, it is unresolved whether this exponent and order of magnitude are optimal; the best known obstruction only gives a lower bound of order p^{k/2}n, arising from K_k-free pseudorandom graphs.