Induced Ramsey Theory in inverse limits

For every finite ordered graph $H$ there is a natural number $k(H)>1$ such that whenever all copies of $H$ in the ordered inverse limit of all finite ordered graphs are partitions to finitely many Borel parts, then there is a (closed) copy of the inverse limit graph in itself whose copies of $H$ meet at most $k(H)$ many parts.

The probability that a random ordered graph on $n$ vertices satisfies $k(H)=1$ tends to 1 as $n$ grows.

Joint work with S. Geschke and S. Huber.