Optimal embeddability threshold for growing-degree random regular graphs

Determine the optimal embeddability threshold as a function of the dimension d and degree Δ for random Δ-regular graphs when Δ tends to infinity, even in the Euclidean setting.

Background

The paper proves a universal non-embeddability result for uniform random Δ-regular graphs when the degree Δ is a fixed constant: with high probability, such a graph cannot be realized as a geometric graph in any normed space of dimension below a universal constant multiple of log n. The authors note that their proof naturally extends to degrees growing as a subpolynomial function of n, but they restrict the main presentation to constant degree.

When Δ tends to infinity, the authors state that the optimal dimension threshold depends jointly on the degree Δ and the ambient dimension d. Even for Euclidean geometric embeddings, the precise threshold is not known, leaving a concrete unresolved problem concerning the dependence on both parameters.