# co.combinatorics – Union closed conjecture induction

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 $$mathcal{A} = {emptyset}$$.

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