# co.combinatorics – The smallest set not covered by the union of a set system

This is a follow-up on this question. Consider the same setup:

Assume that we have a set system $$mathfrak T = {mathcal T_1, mathcal T_2, dots, mathcal T_N }$$ where each $$mathcal T_k$$ is a collection of subsets of $$(n) := {1,dots,n}$$ of the form
$$mathcal T_k = (m_k, M_k) := {T subseteq (n):; m_k subseteq T subseteq M_k }.$$
Moreover, we know that $$mathcal T_k$$s are mutually disjoint, i.e., $$mathcal T_k cap mathcal T_{k’} = emptyset$$ when $$k neq k’$$.

If the union of the sets in $$mathfrak T$$ does not cover $$2^{(n)}$$, can we find the smallest set not covered (or a smallest one if there are multiple) in polynomial-time?

Alternatively, given $$s < n$$, can we find a set of cardinality $$le s$$ that is not covered by $$mathfrak T$$, in time polynomial in $$s$$ (assuming such set exists)?