Archangel Macsika
Show that every simple finite graph has two vertices of the same degree
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Assume the graph G has n vertices. The possibilities for the degree are {0,...,n-1}. However, G cannot have one vertex of degree 0 and one vertex of degree n-1, because these two vertices would need to be adjacent to satisfy degree n-1. Using the pigeonhole principle, we must pick the n degrees (one for each vertex), from a set of n-1 answers (since we cannot have both 0 and n-1). Hence, one degree must be repeated.