SCHWEITZER 2006 Problem 4

let P be a finite set with at least 2 elements. P is a partially ordered and connected set. $p:P^3 \to P$ is a 3-variable, monotone function which satisfies p(x,x,y)=y. Prove that there exists a non-empty subset $I \subset P$ such that $\forall x \in P$ $\forall y \in I$ , we have $p(x, y, y) \in I$ .

[P is connected means that if each element is replaced by vertices and there is an edge between 2 vertices iff the 2 elements can be compared, then the graph is connected. p is monotone means that if $x_1\leq y_1 , x_2\leq y_2 , x_3\leq y_3$ , then $p(x_1,x_2,x_3)\leq p(y_1,y_2,y_3)$ .]