Forbidden Subgraph Bounds for Parallel Repetition and the Density   Hales-Jewett Theorem
  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. 
Jan H\k{a}z{\l}a, Thomas Holenstein, Anup Rao
cs.DM math.CO