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arXiv 2001.088292020-01-23math.CO

High dimensional expansion using zig-zag product

Eyal Karni, Tali Kaufman

备注：: 19 Pages. Some of the results(Conlon's case) were presented in HUJI Combinatorics seminar on June 10th, 2019

英文摘要：

We wish to renew the discussion over recent combinatorial structures that are 3-uniform hypergraph expanders, viewing them in a more general perspective, shedding light on a previously unknown relation to the zig-zag product. We do so by introducing a new structure called triplet structure, that maintains the same local environment around each vertex. The structure is expected to yield, in some cases, a bounded family of hypergraph expanders whose 2-dimensional random walk converges. We have applied the results obtained here to several known constructions, obtaining a better expansion rate than previously known. Namely, we did so in the case of Conlon's construction and the $S=[1,1,0]$ construction by Chapman, Linal and Peled.

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