A graph class is said to have the strong Erdös-Hajnal property (SEH property) if, for every graph in the class, the graph itself or its complement contains a balanced complete bipartite graph of linear size. Recently, we obtain a quantitatively tight result on the SEH property of chordal graphs. In addition, we discuss a colored version of this property for chordal graphs with bounded leafage. In particular, we show that for every pair  of subtree families in a tree with $k$ leaves, there exist subfamilies and with such that either every colorful pair is intersecting or there is no intersecting colorful pair. This is joint work with Minho Cho, Andreas Holmsen, and Jinha Kim.

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