FKS Problem 1 on at most twice the dimension vertices
A Lean-checked proof of FKS Problem 1 for graphs on at most 2d vertices, for every d ≥ 4: every finite simple graph of maximum degree at most d with no connected component isomorphic to K_{d+1} has an injective placement in Euclidean d-space, with every vertex of norm 1/√2 and every edge at distance exactly 1. Non-edges are unconstrained and may also have length 1. The unrestricted question is false at d = 4: Mishra and Senthilkumar (2026) showed that the square of the nine-cycle, on 2d + 1 = 9 vertices, has no such placement. The bound proved here is therefore sharp at d = 4. Neither result…
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