4 problems

Let $K$ be an ultraproduct of the fields $\boldsymbol{Q}_{p_i}$ or $\boldsymbol{F}_{p_i}((t))$ over a non-principal ultrafilter, and let $varphi(x,y)$ be a partitioned formula with…

Let $\mathcal{F}$ be a family of sets in $\mathbb{R}^d$ such that for every non-empty finite subfamily $\mathcal{G}$ and every $i=0,1,\ldots,\lceil d/2\rceil-1$, one has … where…

Let $\mathcal{F}$ be a family of convex sets in $\mathbb{R}^d$, and suppose that an $\alpha$-fraction of the $2d$-tuples of members of $\mathcal{F}$ have intersection of volume at…

Kalai–Meshulam's fractional Helly conjecture. There exists a function $\beta:(0,1)\to(0,1)$ such that, for every $\alpha>0$ and every sufficiently large set system $\mathcal{F}$ in…