SCHWEITZER 1976 Problem 6

Let $0 \leq c \leq 1$ , and let $\eta$ denote the order type of the set of rational numbers. Assume that with every rational number $r$ we associate a Lebesgue-measurable subset $H_r$ of measure $c$ of the interval $[0,1]$ . Prove the existence of a Lebesgue-measurable set $H \subset [0,1]$ of measure $c$ such that for every $x \in H$ the set $$\{r : x \in H_r\ \}$$ contains a subset of type $\eta$ .

M. Laczkovich