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Extendability of $1$-decomposable complexes
Rhea Ghosal, Melody Han, Benjamin Keller, Scarlett Kerr, Justin Liu, SuHo Oh, Ryan Tang, Chloe Weng
https://arxiv.org/abs/2508.04555 …

Extendability of $1$-decomposable complexes
A well-known conjecture of Simon (1994) states that any pure $d$-dimensional shellable complex on $n$ vertices can be extended to $Δ_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex, by attaching one facet at a time while maintaining shellability. The notion of $k$-decomposability for simplicial complexes, which generalizes shellability, was introduced by Provan and Billera (1980). Coleman, Dochtermann, Geist, and Oh (2022) showed that any pure $d$-dimensional $0$-decomposab…