Erdős Problem #44
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Given any finite Sidon set A, can you always extend it to a bigger one of near-(1−ε)√M density? A quantifier-correct construction reaches 1/√2, full density-1 for |A|≤2 (and |A|=3 via a named unpublished proof), and a proof that the obvious repair strategy (patch a perfect difference set) is structurally dead for seeds like Alexeev–Mixon's {1,2,4,8,13}. Round 2: the two |A|=4 patterns that provably embed in no Singer difference set turn out to embed cleanly in a different classical family instead — Singer failure wasn't the wall it looked like. Round 4: the round-3 obstruction sets do embed in Sidon spaces (rebuilt and independently verified, including a real density-value error caught along the way), but a cited nonexistence theorem shows the standard construction family is empty exactly where density-one completions would need to live. Round 5: cubic Bose–Chowla doesn't restore coverage either — a clean, verified structural negative. Round 6: two reasoners working independently converge exactly, and a fact Round 4 left unverified — zero binary critical Sidon spaces at dimension 5 — is now genuinely triple-checked.
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