MathDB
partially ordered, connected set

Source: miklos schweitzer 2006 q4

September 3, 2021
partial orderconnected graphmonotone

Problem Statement

let P be a finite set with at least 2 elements. P is a partially ordered and connected set. p:P3→Pp: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⊂PI \subset P such that ∀x∈P\forall x \in P ∀y∈I\forall y \in I, we have p(x,y,y)∈Ip(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 x1≤y1,x2≤y2,x3≤y3x_1\leq y_1 , x_2\leq y_2 , x_3\leq y_3 , then p(x1,x2,x3)≤p(y1,y2,y3)p(x_1,x_2,x_3)\leq p(y_1,y_2,y_3).]
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