We then focus on the connection between the Frobenius number and arithmetical graphs. Using the Laplacian of an arithmetical graph and a formulation of chip-firing on the vertices of an arithmetical graph, we show results that can be used to find arithmetical graphs whose g-numbers correspond to the Frobenius number of R. We describe how this can be used to quickly calculate upper bounds for the Frobenius number of R.

]]>We give an overview of the concept of the capacity of a hypergraph and survey a few basic results regarding this quantity. Furthermore, we discuss the Lovász number of an undirected graph, which is known to upper bound the capacity of the graph (and in practice appears to be the best such general purpose bound).

We then elaborate on some attempted generalizations/modifications of the Lovász number to undirected hypergraphs that we have tried. It is not currently known whether these attempted generalizations/modifications upper bound the capacity of arbitrary hypergraphs.

An important method for proving lower bounds on hypergraph capacity is to exhibit a large independent set in a strong power of the hypergraph. We examine methods for this and show a barrier to attempts to usefully generalize certain of these methods to hypergraphs.

We then look at cap sets: independent sets in powers of a certain hypergraph. We examine certain structural properties of them with the hope of finding ones that allow us to prove upper bounds on their size.

Finally, we consider two interesting generalizations of capacity and use one of them to formulate several conjectures about connections between cap sets and sunflower-free sets.

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