The Milliken–Taylor theorem is a result in the field of graph theory, particularly concerning the coloring of graphs. It provides a criterion for determining the chromatic number of certain types of graphs, specifically those that are constructed from the edges of a complete graph.
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The Milliken–Taylor theorem gives monochromatic systems of separated block sums with a fixed compressed coefficient vector, such as . Systems associated with two nonproportional compressed coefficient vectors need not have the same color: in particular, there is a finite coloring for which a finite-sums set and the Milliken–Taylor system cannot be monochromatic together.