TITLE: Vertices of a complete graph $K_6$ are painted in black and white: find a subgraph $K_3$ with vertices of the same color
QUESTION [0 upvotes]: Vertices of a complete graph $K_6$ are painted in black and white. Show that it contains a subgraph $K_3 \subset K_6$ with vertices of same color.
I am quite newbie at discrete maths. So, any help would be appreciated. Thanks.

REPLY [1 votes]: there must be at least 3 black vertices or at least 3 white vertices. Any three vertices in a complete graph combine to a $K_3$. -- Leen Droogendijk

Note that the above remains true if $K_6$ is replaced with $K_5$.
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