How to find an example of a union-closed family with two given properties

Question

Consider a union-closed family $\mathcal{F}$ of $n=\vert \mathcal{F} \vert$ sets, $n$ odd, and its family $\mathit{J}(\mathcal{F})$ of $m = \vert\mathit{J}(\mathcal{F})\vert$ basis sets (or $\cup$-irreducible sets, also called generators), and define $\mathcal{F_a} = \{A \in \mathcal{F}: a \in A \}$, $\mathit{J_a}(\mathcal{F}) = \{A \in \mathit{J}(\mathcal{F}) : a \in A \}$.

It is easy to prove by contradiction that if $\exists a$ such that $\vert\mathit{J_a}(\mathcal{F})\vert \ge m-\Bigl\lfloor\log_2{\frac{n+1}{2}}\Bigr\rfloor$, then $\vert \mathcal{F_a} \vert \ge \frac{n}{2}$.

I have tried the two examples in "The journey of the union-closed sets conjecture" by Bruhn and Schaudt on section 6.3 "Limits of averaging" and both seem to satisfy the above requirement.

I understand it might not be easy, but any idea on how to construct an example of a union-closed family with average set size less than half of the universe and such that $\forall a$ $\vert\mathit{J_a}(\mathcal{F})\vert \lt m-\Bigl\lfloor\log_2{\frac{n+1}{2}}\Bigr\rfloor$?

Answer 1

Not a good example, but it might hint for a better one.

Take two integers $N=2k+1,M$. Let the basis sets be $\{x,x+1,...,x+k\},1\leq x\leq N$, taken modulo $N$ (identify $N\equiv 0$) and $\{1,2,...,N,a_1,a_2,...,a_M\}$.

The union-closed family $\mathcal{F}$ contains all set of the form $\{x,x+1,...,x+l\},1\leq x\leq N,k\leq l\leq N-1$ and $\{1,2,...,N,a_1,a_2,...,a_M\}$. So it's easy to see that $m=N+1,n=\frac{N(N-1)}{2}+2,J_i(\mathcal{F})=\frac{N+1}{2},J_{a_j}(\mathcal{F})=1$. The average set size is $f(N)+\frac{M}{\frac{N(N-1)}{2}+2}$ ($f(N)$ is quite complicative to write, but it's independent of $M$).

Now we choosing $N,M$ such that $\frac{N-1}{2}$ is odd, $log_2(\frac{N(N-1)}{2}+2)<\frac{N+1}{2}, f(N)+\frac{M}{\frac{N(N-1)}{2}+2}<\frac{N+M}{2}$ then $\mathcal{F}$ is the union-close family that we want.