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Recently the breakthrough method of hypergraph containers, developedindependently by Balogh, Morris, and Samotij as well as Saxton and Thomason,has been used to study sparse random analogs of a variety of classical problemsfrom combinatorics and number theory. The previously known proofs of thecontainers theorem use the so-called scythe algorithm---an iterative procedurethat runs through the vertices of the hypergraph. (Saxton and Thomason havealso proposed an alternative, randomized construction in the case of simplehypergraphs.) Here we present the first known deterministic proof of thecontainers theorem that is not algorithmic, i.e., it does not involve aniterative process. Our proof is less than 4 pages long while being entirelyself-contained and conceptually transparent. Although our proof is completelyelementary, it was inspired by considering hypergraphs in the setting ofnonstandard analysis, where there is a notion of dimension capturing thelogarithmic rate of growth of finite sets. Before presenting the proof in fulldetail, we include a one-page informal outline that refers to this notion ofdimension and summarizes the essence of the argument.

A short nonalgorithmic proof of the containers theorem for hypergraphs