Sharp threshold for reconstructing points on the line

TL;DR

This paper proves that for a random graph on real points with edge probability slightly above 1/n, a large reconstructible subset exists in the largest component, confirming a conjecture about the threshold for reconstructibility.

Contribution

It establishes that whp, a reconstructible subset of nearly the entire largest component exists for edge probability above 1/n, strengthening previous conjectures.

Findings

Existence of a large reconstructible subset in the largest component for p=(1+ε)/n

The size of this subset is asymptotically the size of the component

Results extend to ε(n) satisfying ε=ω(1/ln n)

Abstract

For a set of $n$ points $V \subseteq \mathbb{R}$ let $G(V, p)$ be the random graph on $V$ where each possible edge is present independently with probability $p$. We call a subset $U \subseteq V$ {\emph {reconstructible}} if every injection $\varphi:V\to \mathbb{R}$ that preserves the distances along the edges of $G(V, p)$ also preserves all pairwise distances in $U$. How large is the size $\mathsf{R}$ of a largest reconstructible subset? Gir\~ao, Illingworth, Michel, Powierski and Scott conjectured that the answer is linear whp when $p = (1+\varepsilon)/n$ for every $\varepsilon > 0$. In this paper, we show that for every $\varepsilon>0$ whp there exists a reconstructible subset $U$ of the largest component $\mathcal{C}$ of the 2-core satisfying $|U| = |V(\mathcal{C})|(1-o(1))$, proving a stronger form of the conjecture. The bound is asymptotically best possible, since for $V \subseteq \mathbb{R}$ linearly independent over $\mathbb{Q}$ it is straightforward to verify that $\mathsf{R} \leq \max(2, |V(\mathcal{C})|)$. Furthermore, we extend these results to every $\varepsilon:= \varepsilon(n)$ satisfying $\varepsilon = \omega(1/\ln n)$.