let P be a finite set with at least 2 elements. P is a partially ordered and connected set. p:P3→P is a 3-variable, monotone function which satisfies p(x,x,y)=y. Prove that there exists a non-empty subset I⊂P such that ∀x∈P ∀y∈I, we have p(x,y,y)∈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 x1≤y1,x2≤y2,x3≤y3 , then p(x1,x2,x3)≤p(y1,y2,y3).] partial orderconnected graphmonotone