Asymptotically attaining the Moore bound

For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. It is shown that $$\lim_{d \to \infty}\frac{n_k(d)}{d^k} = 1$$ for every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter. Also proved a similar lower bound on the edge-variant of the problem, and a tight asymptotic for the bipartite variant of the edge problem.

Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.