On High-Dimensional Acyclic Tournaments

Abstract

We study a high-dimensional analog for the notion of an acyclic (aka transitive) tournament. We give upper and lower bounds on the number of d-dimensional n-vertex acyclic tournaments. In addition, we prove that every n-vertex d-dimensional tournament contains an acyclic subtournament of Ω(log 1/d n) vertices and the bound is tight.This statement for tournaments (i.e., the case d = 1) is a well-known fact. We indicate a connection between acyclic high-dimensional tournaments and Ramsey numbers of hypergraphs. We investigate as well the inter-relations among various other notions of acyclicity in high-dimensional tournaments. These include combinatorial, geometric and topological concepts.

A universal exponent for homeomorphs

“…A number of natural extremal geometric problems arise when we view an r-uniform hypergraph as an (r − 1)-dimensional simplicial complex (by identifying edges with facets). Questions of this nature arise in the high-dimensional combinatorics programme of Linial [12,10], and have also been raised by Gowers [5]; for a sample of some recent results in this programme, see [4,14,13,11]. In this paper, we study the Turán problem for 2-complexes, or equivalently, the topological Turán problem for 3-graphs.…”

Cycles in Oriented 3-Graphs

“…Following [4], the incidence matrix of an oriented 3-graph G is an n 2 × |E| matrix A whose rows and columns correspond to all 2-sets [n] (2) …”

“…Linial and Morgenstern [4] introduced a notion of 'cycle' in an oriented 3-graph. Roughly speaking (we will give a precise definition at the start of Sect.…”

“…Linial and Morgenstern [4] introduced a notion of 'cycle' in an oriented 3-graph. Roughly speaking (we will give a precise definition at the start of Section 2), a cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set.…”

Polynomial to exponential transition in Ramsey theory

“…Note that so includes classical Ramsey numbers. In addition, the case was investigated in relation to the Erdős–Szekeres theorem and Ramsey numbers of ordered tight paths as well as to high‐dimensional tournaments by several researchers [9, 10, 15, 26–28]; the very special case has connections to quasirandom hypergraph constructions [3, 21, 24, 25].…”