I would like an example showing that one of the most basic induction approaches to the union-closed conjecture fails. If, for any union-closed family $mathcal{A}$ of subsets of a finite set $X$, there is some $x in X$ such that each $y in X$ has $|{A in mathcal{A} : A ni y text{ and } A ni x}| ge frac{1}{2}|{A in mathcal{A} : A ni x}|$, then we can merely use induction applied to the union-closed family ${A in mathcal{A} : A not ni x}$ to get some $y in X$ in at least half of the sets of ${A in mathcal{A} : A not ni x}$, and by our choice of $x$, we then see that $y$ is in at least half the sets of $mathcal{A}$.

I have to think that there is a known example showing this approach doesn’t work, i.e., there is an $mathcal{A}$ with no such $x$. But I couldn’t think of an example. So,:

Give an example of a finite set $X$ and a union-closed family $mathcal{A} subseteq mathcal{P}(X)$ such that, for each $x in X$, there is some $y in X$ with $$|{A in mathcal{A} : A ni y text{ and } A ni x}| < frac{1}{2}|{A in mathcal{A} : A ni x}|.$$ (Or prove the union-closed conjecture!)

I avoid degenerate cases, like $X = emptyset$, $mathcal{A} = emptyset$, or $mathcal{A} = {emptyset}$.