In a game, all points in the (x, y) plane with coordinates obeying x , y ∈ Z are labelled as belonging to one of three players, i.e., either to Alice, Bob or Carol.Show that one of the players will possess four points whose vertices form a rectangle.Here is the problem I've been thinking for days. It seems easy since the coordinates are unlimited. But it is also hard to find pigeonholes I want to divide.I've encountered several problems like finding a parallelogram on a n × n chessboard with 2 n pawns. It is relatively simpler to me.

Question

In a game, all points in the (x, y) plane with coordinates obeying   x  ,  y  ∈      Z   are labelled as belonging to one of three players, i.e., either to Alice, Bob or Carol.Show that one of the players will possess four points whose vertices form a rectangle.Here is the problem I&#039;ve been thinking for days. It seems easy since the coordinates are unlimited. But it is also hard to find pigeonholes I want to divide.I&#039;ve encountered several problems like finding a parallelogram on a   n  ×  n chessboard with   2  n pawns. It is relatively simpler to me.

Haggar72 · Accepted Answer

Consider the points   ⟨  0  ,  n  ⟩,   ⟨  1  ,  n  ⟩,   ⟨  2  ,  n  ⟩, and   ⟨  3  ,  n  ⟩ for   n  =  0  ,  1  ,  2  ,  …  . Imagine that each point is labelled   A,   B, or   C. Thus, one row might be labelled   A  B  C  A, another might be labelled   B  C  C  B, and so on.1. How many different labellings of a row are possible?2. Why would you like to find two rows with the same labelling?3. How many rows does it take to guarantee that you have two rows with the same labelling?

In a game, all points in the (x, y) plane with coordinates obeying x , y ∈<!-- ∈

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