  We study a special kind of bounds (so called forbidden subgraph bounds, cf.
Feige, Verbitsky '02) for parallel repetition of multi-prover games.
  First, we show that forbidden subgraph upper bounds for $r \ge 3$ provers
imply the same bounds for the density Hales-Jewett theorem for alphabet of size
$r$. As a consequence, this yields a new family of games with slow decrease in
the parallel repetition value.
  Second, we introduce a new technique for proving exponential forbidden
subgraph upper bounds and explore its power and limitations. In particular, we
obtain exponential upper bounds for two-prover games with question graphs of
treewidth at most two and show that our method cannot give exponential bounds
for all two-prover graphs.

Discrete Mathematics;Combinatorics