SCHWEITZER 1992 Problem 4

show there exist positive constants $c_1$ and $c_2$ such that for any $n\geq 3$ , whenever $T_1$ and $T_2$ are two trees on the set of vertices $X = \{1, 2, ..., n\}$ , there exists a function $f : X \to \{-1, +1\}$ for which $$\bigg | \sum_ {x \in P} f (x) \bigg | <c_1 \log n$$ for any path P that is a subgraph of $T_1$ or $T_2$ , but with an upper bound $c_2 \log n / \log \log n$ the statement is no longer true.