SCHWEITZER 2015 Problem 3

Let ${A}$ be a finite set and ${\rightarrow}$ be a binary relation on it such that for any ${a,b,c \in A}$ , if ${a\neq b}, {a \rightarrow c}$ and ${b \rightarrow c}$ then either ${a \rightarrow b}$ or ${b \rightarrow a}$ (or possibly both). Let ${B, B \subset A}$ be minimal with the property: for any ${a \in A \setminus B}$ there exists ${b \in B}$ , such that either ${a \rightarrow b}$ or ${b \rightarrow a}$ (or possibly both). Supposing that ${A}$ has at most ${k}$ elements that are pairwise not in relation ${\rightarrow}$ , prove that ${B}$ has at most ${k}$ elements.