2 problems

Simmons and Su's conjecture. There must exist $k$ adjacent vertices in the triangulation whose labels are $\{\omega^j m:1\leq j\leq k\}$ for some fixed $m$.

Borsuk–Ulam conjecture. With high probability, for every continuous map $f\colon\widetilde Z\to\mathbb{R}^{2d+1}$, there exists an $x\in|\widetilde Z|$ such that