  Let K_4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai,
Erd\H{o}s, Koml\'os, and Szemer\'edi (1981) asked if there is a function
\omega(d) tending to infinity such that every 3-uniform, K_4-free hypergraph N
vertices and average degree d has independence number at least \omega(d)
N/d^{1/2}. We answer this question by constructing a 3-uniform, K_4-free
hypergraph with independence number at most 2N/d^{1/2}. We also provide
counterexamples to several related conjectures and improve the lower bound of
some hypergraph Ramsey numbers.
