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Ramsey theorem for 2-colored graph

Task number: 4014

Prove that for every k∈N there exists n∈N such that for every graph G=(V,E) with at least n vertices and for each its edge coloring c:E→{1,2} there is U⊆V of size at least k such that all edges of the induced subgraph G[U] have the same color.

  • Solution

    We will complete the non-edges and color them arbitrarily. Then we apply the classical Ramsey theorem for two colors in the complete graph.

Difficulty level: Easy task (using definitions and simple reasoning)
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