MathDB
Miklós Schweitzer 1984- Problem 1

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September 4, 2016
college contests

Problem Statement

1. Let kk be an arbitrary cardinality. Show that there exists a tournament Tk=(Vn,En)T_k = (V_n , E_n) such that for any coloring f:En→kf: E_n \to k of the edge set EnE_n, there are three different vertices x0,x1,x2∈Vnx_0 , x_1 , x_2 \in V_n such that
x0x1,x1x2,x2x0∈Enx_0 x_1 , x_1 x_2 , x_2 x_0 \in E_n
and
∣{f(x0x1),f(x1x2),f(x2x0)}∣≤2\left | \{ f(x_0 x_1), f(x_1 x_2), f(x_2 x_0)\} \right |\leq 2
(A tounament is a directed graph such that for any vertices x,y∈Vn,x≠yx, y \in V_n, x \neq y exactly one of the relations xy∈Enxy \in E_n holds.) (C.19) [A. Hajnal]
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