CIIM 2012 Problem 5

Let $D=\{0,1,\dots,9\}$ . A direction function for $D$ is a function $f:D \times D \to \{0,1\}.$ A real $r\in [0,1]$ is compatible with $f$ if it can be written in the form $$r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}$$ with $d_j \in D$ and $f(d_j,d_{j+1})=1$ for every positive integer $j$ .

Determine the least integer $k$ such that for any direction fuction $f$ , if there are $k$ compatible reals with $f$ then there are infinite reals compatible with $f$ .