Question 78
Let $A$ and $B$ be sets with $2011^2$ and $2010$ elements, respectively. Show that there is a function $f: A \times A \to B$ satisfying the condition $f(x,y) = f(y,x)$ for all $(x,y) \in A \times A$ such that for every function $g: A \to B$ there exists $(a_1, a_2) \in A \times A$ with $g(a_1) = f(a_1, a_2) = g(a_2)$ and $a_1 \neq a_2$.