Completing shellings with $d-2$ extra vertices

Source abstract

Simon's conjecture (1994) asserts that any pure $d$-dimensional shellable complex on $n$ vertices can be extended to the $d$-skeleton of the simplex on $n$ vertices, one facet at a time, while maintaining shellability. Bolognini and Sentinelli (2026) recently disproved it for every $d \geq 3$. We show that the conjecture becomes true once $d-2$ new vertices are allowed, and that the same holds with $k$-decomposability in place of shellability for any $k \geq 1$. We also show that the number $d-2$ cannot be lowered. Inflating the counterexample of Bolognini and Sentinelli gives, for every $d \geq 3$, a $1$-decomposable complex that cannot be extended in this way with only $d-3$ new vertices, even while merely maintaining shellability.