Let $K$ be a simplicial complex with $n$ vertices and $n-t$ facets, where $t \geq 3$.


Is it true that the $(n-3-j)$th reduced homology (with coefficients in a field) of $K$ vanishes for $0 \leq j \leq t-2\,$?



Vanishing homology of simplicial complexes with few facets
By the nerve lemma, your complex is homotopy equivalent to a complex with $n-t$ vertices and therefore has trivial homology in degrees greater than $n-t-2$


