Borsuk–Ulam conjecture for random complexes with free involution

Let $d\geq 1$ and let $\alpha$ satisfy

$$\frac{1}{d+1}<\alpha<\frac{1}{d}.$$

Let $\widetilde Z\sim\widetilde Z(2n,n^{-\alpha})$ be a random simplicial complex on $2n$ vertices with free involution $\iota\colon\widetilde Z\to\widetilde Z$.

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

$$f(x)=f(\iota(x)).$$

The preceding theorem proves the analogous assertion with target $\mathbb{R}^{d+1}$ in this probability regime, using that $\widetilde Z$ is at least $d$-connected with high probability. The conjecture reflects the observation that the connectivity over the rationals is expected to be about twice as large, but the stated stronger Borsuk–Ulam property remains open.