Erdős–Purdy–Agarwal–Sharir conjecture on congruent simplices

Let $d\geq 4$, let $k\leq d$, and let $\mathrm{C}(d,k,n)$ denote the maximum, over all $k$-vertex simplices $S$ and all $n$-element sets $P\subseteq\mathbb{R}^d$, of the number of occurrences of a congruent copy of $S$ in $P$. Erdős–Purdy–Agarwal–Sharir conjecture.

For $d\geq 4$ and $k\leq d$ we have

$$\mathrm{C}(d,k,n)=O_d(\min\{n^k,n^{d/2}\})$$

if $d$ is even, and

$$\mathrm{C}(d,k,n)=O_d(\min\{n^k,n^{d/2-1/6}\})$$

if $d$ is odd.

This conjecture concerns the higher-dimensional generalisation of the unit-distance problem, where one counts congruent simplices rather than pairs at a fixed distance. The paper develops upper bounds using classical cutting techniques, while the conjectured estimates remain unresolved in general.