Frankl-Kupavskii-Swanepoel Theorem 3: fewer than C(d+2,2) edges embeds in R^d
For every integer d ≥ 4, every finite simple graph with fewer than C(d + 2, 2) edges has an injective placement in Euclidean d-space under which every edge has length 1, and the complete graph K_{d+2} has exactly C(d + 2, 2) edges and has no such placement. Equivalently, f(d) = C(d + 2, 2), where f(d) is the least number of edges in a finite simple graph with no unit-distance representation in R^d. Non-edges may also lie at distance 1. Finite simple graphs are represented as SimpleGraph (Fin n), including the empty graph. The proof is Frankl–Kupavskii–Swanepoel Theorem 3, already proved in…
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