Rabdos Math Bench

For a finite family $\mathcal{F}$ of distinct $k$-element sets, let $d(\mathcal{F})$ be the collection of all $(k-1)$-element sets contained in at least one member of $\mathcal{F}$, and write $D(\mathcal{F}) = |d(\mathcal{F})|$. Let $M(k)$ be the largest positive integer that is not equal to $D(\mathcal{F})$ for any finite family $\mathcal{F}$ of distinct $k$-element sets. For fixed $k$ and $t$, let $L_k(t)$ be the smallest integer $L$ such that every integer from $L$ through $tk$ is equal to $D(\mathcal{F})$ for some $t$-member family $\mathcal{F}$ of distinct $k$-element sets. Let $\tau(k)$ be the largest $t$ in $\{1, \ldots, k+1\}$ with $L_k(t) \ge tk - k + 2$.

Work over $\mathbb F_2$. Let $A=\mathbb F_2[\xi_1,\xi_2,\ldots]$, with $|\xi_i|=2^i-1$ and coproduct

A closed polygonal curve in the plane is called generic if its vertices are distinct, no three vertices are collinear, no vertex lies in the relative interior of a nonincident edge, and no three edges have a common point. Thus every intersection between nonadjacent edges is a transverse double crossing.