  We analyze the asymptotic extremal growth rate of the Betti numbers of clique
complexes of graphs on n vertices not containing a fixed forbidden induced
subgraph H.
  In particular, we prove a theorem of the alternative: for any H the growth
rate achieves exactly one of five possible exponentials, that is, independent
of the field of coefficients, the nth root of the maximal total Betti number
over n-vertex graphs with no induced copy of H has a limit, as n tends to
infinity, and, ranging over all H, exactly five different limits are attained.
  For the interesting case where H is the 4-cycle, the above limit is 1, and we
prove a slightly superpolynomial upper bound.
