Upper bound theorem for odd-dimensional flag triangulations of manifolds

TL;DR

This paper establishes that for odd-dimensional flag triangulations of manifolds with many vertices, a specific balanced join of cycles uniquely maximizes various combinatorial vectors, using extremal graph theory techniques.

Contribution

It proves a new extremal upper bound theorem for flag triangulations of odd-dimensional manifolds, identifying the unique maximizer as a balanced join of cycles.

Findings

Balanced join of r cycles maximizes f-, h-, g-, and gamma-vectors

Unique maximizer among all flag triangulations of odd-dimensional manifolds

Uses extremal graph theory methods in topological combinatorics

Abstract

We prove that among all flag triangulations of manifolds of odd dimension 2r-1 with sufficiently many vertices the unique maximizer of the entries of the f-, h-, g- and gamma-vector is the balanced join of r cycles. Our proof uses methods from extremal graph theory.