Would appreciate any insight about the following regarding set covers:
Begin with a universe set $X = \{x_1,x_2,...,x_n\}$ and a set $S=\{s_1, s_2,...,s_p\}$ such that each $s_i \subseteq X$ and $\bigcup S = X$. Consider the task of finding all inclusion-minimal covers of $X$. (Covers are inclusion-minimal if the removal of any subset $s_i$ from the cover destroys its coverage property.)
Given the set of inclusion-minimal covers $C=\{c_1, c_2, ... c_m\}$, what is an upper bound for $\max|c_i|$, that is, the largest inclusion-minimal cover? It seems fairly straightforward to show that one upper bound is $|X|$ (that is, an inclusion-minimal cover will contain at most the same number of subsets as there are elements in $X$ itself).
But can a better bound be found?
Additional information: No restrictions are assumed for $S$. The bound would ideally be expressed in terms of one or more properties of $S$ (in whatever form those might take).
A conjecture: If $\min |s_i| = k$, then $\max |c_i|$ (the size of the largest inclusion-minimal cover) is bounded above by $|X| - k + 1$. This is based upon two observations:
(1) In constructing any inclusion-minimal cover, we start with any element from $S$ (call it $s_{1'}$). As $s_{1'}$ covers a particular subset of elements from $X$, the next added cover element (call it $s_{2'}$) must cover at least one new element from $X$ not covered by $s_{1'}$.
(2) If $\min |s_i| = k$, then we can choose the first element $s_{1'}$ in our cover to have size $k$. This leaves $|X| - k$ elements in $X$ left uncovered. By the first observation, each additional $s_i$ must cover at least one new element in $X$, so we can add at most $|X| - k$ additional elements to the cover, leading to a cover size at most $|X| - k + 1$.
As noted in the comments, this type of bound might not be significantly smaller than $|X|$ itself. However, I am hoping to use this in a combinatorial context, so any savings helps. Even smaller bounds would still be useful.
