Finding and counting cliques and independent sets in r-uniform hypergraphs

We present a matrix multiplication based algorithm for counting the number of (induced) occurrences of a fixed r-uniform hypergraph in a larger hypergraph. In many cases, the running time is better than that of the naïve algorithm. We also present several useful applications of the algorithm, such as determining the dominant color among monochromatic simplices in a red-blue edge-colored hypergraph, approximating the number of independent simplices in a random hypergraph, and counting induced occurrences of a given 3-uniform k-vertex hypergraph in a larger k-clique free hypergraph.