Expansion and Minors in Hypergraphs
Uli Wagner
ETH Zürich
Let X be a 3-uniform hypergraph, or in other words, a 2-dimensional simplicial
complex.
We discuss several notions of what it means for X to be expanding, with a
particular focus on a generalization of edge expansion that arose in recent work of
Gromov (and, independently, of Linial and Meshulam).
We also discuss a new notion of minors for hypergraphs/simplicial complexes,
called homological minors. The definition is motivated by extremal problems relating
the combinatorics of a hypergraph (the number of hyperedges/faces) to its topological properties such as (non)embeddability in Euclidean spaces or the existence of
Tverberg-type partitions.
We then relate the two notions and show that every ε-expanding 2-dimensional
hypergraph with C triangles incident to every edge (on average) contains a large
2-dimensional complete minor on t vertices (provided ε and C are sufficiently large
in terms of t). (While this statement is what one would expect in view of analogous statements for graphs, the arguments in higher-dimensions are rather different.) As a corollary, one also obtains asymptotically sharp bounds for the threshold
probabilities at which random complexes stop being embeddable and start having
Tverberg-type substructures. All of these happen at p = Θ(1/n) (with a constant
depending on the precise context).