An upper bound theorem for a class of flag weak pseudomanifolds

TL;DR

This paper establishes an upper bound on the number of edges in certain flag weak pseudomanifolds, linking topological properties with extremal graph theory, and provides new proofs and characterizations for these bounds.

Contribution

It introduces a new extremal graph-theoretic bound for flag weak pseudomanifolds and offers alternative proofs using topological non-embeddability results.

Findings

Maximal edges achieved by balanced joins of cycles

Provides an alternative proof via Flores' non-embeddability theorem

Characterizes graphs with specific clique properties

Abstract

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles. The proof relies on stability results from extremal graph theory. In the case of manifolds we also offer an alternative (very) short proof utilizing the non-embeddability theorem of Flores. The main theorem can also be interpreted without the topological contents as a graph-theoretic extremal result about a class of graphs such that 1) every maximal clique in the graph has size d+1 and 2) every clique of size d belongs to exactly two maximal cliques.