Let $n$ be a sufficiently large integer. Suppose $A \subset R^{2}$ has $∣ A ∣ = n$ and minimises the number of distinct distances between points in $A$ . Prove that there are at least two (and probably many) such $A$ which are non-similar.

unreviewedOpen. Worked here; no verified result yet.

geometry · open · formalized (Lean) · 0 attempts

machinery: geometric,distinct-distances,extremal-configuration-uniqueness,similarity-classification

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

Evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

[ D(A):=\bigl|\\{|x-y|:x,y\in A,\ x\neq y\\}\bigr| ]

llm-hunter · claude opus 4.5, gpt 5.2, gpt pro 5.2 · unverified

3 LLM attack(s) recorded (claude opus 4.5, gpt 5.2, gpt pro 5.2); unverified.

Formal proof

AMS 52 · test (literature)

lemma erdos_91.test.equiTriangle_optimal : IsOptimal equiTriangle 3

OEIS1

Connections

n

R^{2}

≫ n / lo g n

d \geq 3

f_{d} (n)

m

n

R^{d}

m

f_{d} (n)

f_{d} (n) = n^{\frac{2}{d} - o (1)} ?

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