Let $ G$ be a finite set, and let $ \mathcal A\ni G$ be a subset of $ \mathcal P(G)$ .

For any $ A\in \mathcal A$ we call $ cov(A)\in \mathcal A$ the smallest element of $ \mathcal A$ such that $ A\subset cov(A)$

(i.e if $ A\subset B\subset cov(A)$ then $ B\notin \mathcal A$ ).

We say that $ a\in G$ is full in $ Y\in \mathcal A$ , and write $ F_{\mathcal A}(a,Y)$ if $ a\in X$ for any $ X$ such that $ cov(X)=Y$ ,

and we define $ F_{\mathcal A}(a,.)=:\left\{Y\in \mathcal A,\,\, F_{\mathcal A}(a,Y)\right\}$

Question

Does there exists a constant $ C<1$ such that for any finite set $ G$ and any $ \mathcal A\in \mathcal P(G)$ such that $ G\in \mathcal A$ , the inequality

$ |F_{\mathcal A}(a,.)|/|\mathcal A|\leq C$

holds?

For example if $ G_1=\left\{a,b,c,d\right\}$ , and $ \mathcal A_1= \left\{\left\{a\right\},\left\{a,b\right\},\left\{a,b,c\right\},G_1\right\}$ then, for any $ x\in G_1$ and $ Y\in \mathcal A_1$ , $ x$ is full in $ Y$ iff $ Y\neq \left\{x\right\}$

Other example : if $ \mathcal A_2$ is a finite down-set with ground set $ G_2$ , than $ F_{\mathcal A_2}(x,.)$ is empty for any $ x\in G_2$ .

One can show that if $ \mathcal A’$ is closed under intersection with ground set $ G’$ then for ANY $ x\in G’$ we have :

$ |\left\{A\in \mathcal A’, x\in A\right\}|<(|\mathcal A’|+|F_{\mathcal A’}(x,.)|)/2$

This last remark shows the link between the question and the Frankl Conjecture https://en.wikipedia.org/wiki/Union-closed_sets_conjecture