Intersection patterns in spaces with a forbidden homological minor

Abstract In this paper we study generalizations of classical results on intersection patterns of set systems in R d $\mathbb {R}^d$ double struck upper R Superscript d , such as the fractional Helly theorem or the ( p , q ) $(p,q)$ left parenthesis p comma q right parenthesis -theorem, in the setting of arbitrary triangulable spaces with a forbidden homological minor. Given a simplicial complex K and an integer b , we say that a family F $\mathcal {F}$ script upper F of subcomplexes of some simplicial complex X is a (K,b)-free cover if (i) K is a forbidden homological minor of X , and (ii) the j th reduced Betti number β ~ j ( ⋂ S ∈ G S , Z 2 ) $\tilde {\beta }_j(\bigcap _{S\in {\mathcal {G}}}S,\mathbb {Z}_2)$ ModifyingAbove beta With tilde Subscript j Baseline left parenthesis intersection Underscript upper S element of script upper G Endscripts upper S comma double struck upper Z 2 right parenthesis is strictly less than b for all 0 ≤ j < dim ⁡ K $0\leq j < \dim K$ 0 less than or equals j less than dimension upper K and all nonempty subfamilies G ⊆ F $\mathcal {G}\subseteq \mathcal {F}$