Krivelevich–Lew–Michaeli conjecture on maximal rigid dimension in G(n,p)

Establish that for any edge probability p(n) = ω(log n / n) and any fixed ε > 0, the Erdős–Rényi random graph G(n,p) is asymptotically almost surely d-rigid for every integer d < (1 − ε) n (1 − √(1 − p)).

Background

The paper investigates the largest dimension d for which an Erdős–Rényi random graph G(n,p) is generically rigid. For fixed d, rigidity coincides with the minimum degree threshold, but for p well above the connectivity threshold, the number of edges becomes a bottleneck. The cited conjecture asserts that this edge-count barrier is sharp across the entire range p = ω(log n / n).

The authors confirm this conjecture in a specific regime (p_c ≤ p = o(n^{−1/2})) by showing d(n,p) ≈ (1/2)np, but the full conjecture remains unresolved beyond that range.