Determining a Points Configuration from a Subset of the Pairwise Distances

We study rigidity without assuming general position. Given $n$ distinct labelled points and a set $\mathcal{P}\subseteq \binom{[n]}{2}$ of revealed pairs, we ask when the corresponding distances determine the configuration up to isometry. On the line, we prove an extremal result: if $|\mathcal{P}|=\Omega(n^{3/2})$, then there is an induced globally rigid subgraph on $\Omega(|\mathcal{P}|/n)$ vertices. In other words, any dense enough graph will contain a subset of labels whose locations can be determined from their distances up to isometry. To prove this, we establish a graph-theoretic result, which may be of independent interest: a dense graph in which every non-edge has few common neighbours contains a clique of size $\Omega(|E|/n)$. We also study random revealed pairs. For every labelled configuration $V$ of distinct points in $\mathbb{R}$, if each pair is revealed independently with probability $p=C\ln n/n$, where $C>1$, then the revealed distances determine $V$ w.h.p. We prove a similar result for $d\ge1$ under the mild non-degeneracy assumption that every subcollection of more than $\tau n$ points of $V\subseteq\mathbb R^d$ affinely spans $\mathbb R^d$, for some fixed $0<\tau<1$. In this case, every $C>1/(1-\tau)$ suffices. The same ideas also settle the weak-threshold form of a conjecture of Gir\~ao et al. for a giant reconstructable component, and substantially improve in this direction the work of Barnes et al. establishing such a component for $p>n^{-2/(d+4)}$.