Planar Maximally Filtered Graphs (PMFG) are an important tool for filtering the most relevant information from correlation based networks such as stock market networks. One of the main characteristics of a PMFG is the number of its 3-and 4-cliques. Recently in a few high impact papers it was stated that, based on heuristic evidence, the maximum number of 3-and 4-cliques that can exist in a PMFG with n vertices is 3n − 8 and n − 4 respectively. In this paper, we prove that this is indeed the case.
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