Monochromatic Clique Decompositions of Graphs

Let G be a graph whose edges are colored with k colors, and [Formula: see text] be a k‐tuple of graphs. A monochromatic [Formula: see text] ‐decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic copy of [Formula: see text] in color i, for some [Formula: see text]. Let [Formula: see text] be the smallest number ϕ, such that, for every order‐n graph and every k‐edge‐coloring, there is a monochromatic [Formula: see text] ‐decomposition with at most ϕ elements. Extending the previous results of Liu and Sousa [Monochromatic [Formula: see text] ‐decompositions of graphs, J Graph Theory 76 (2014), 89–100], we solve this problem when each graph in [Formula: see text] is a clique and [Formula: see text] is sufficiently large.