Kalai–Sarkaria face-number conjecture for embeddable complexes

Let $d,r \geq 1$ satisfy $d \leq r \leq 2d$, and let $n \geq r+1$. Let $K$ be a $d$-dimensional simplicial complex on $n$ vertices that embeds into $\mathbb{R}^{r}$. Write $f_k(K)$ for the number of $k$-simplices of $K$, and let $C_{r+1}(n)$ be the cyclic $(r+1)$-polytope with $n$ vertices.

Kalai–Sarkaria face-number conjecture. The complex $K$ has at most as many $d$-simplices as $\partial C_{r+1}(n)$, and

$$f_k(K) \leq f_k(C_{r+1}(n))$$

for all $k \in \{1,\ldots,d\}$. Consequently,

$$\max\left\{f_d(K) \;\middle|\; \dim(K)=d,\, |V(K)|=n,\, ||K|| \hookrightarrow \mathbb{R}^{r} \right\} \leq f_d(C_{r+1}(n))=O(n^{\lceil\frac{r}{2}\rceil}).$$

This is presented as an immediate consequence of the Kalai–Sarkaria shifting conjecture and gives the expected upper bound on the size of embeddable complexes in the range $d \leq r \leq 2d$.