KöMaL 2012 A573

Let $D$={0,1,2,...,9} be the set of decimal digits, and let $R \subset D$×$D$be a set of ordered pairs of digits. An infinite sequence ($a_1$,$a_2$,$a_3$,...) of digits is said to be compatible with R if ($a_j$,$a_{j+1}$)$\in R$ for all positive integer $j$. Determine the smallest positive integer $K$ with the property that if an arbitrary set $R \subset D$×$D$ is compatible with at least $K$ distinct digit sequences then $R$ is compatible with infinitely many digit sequences.