{"schema":"vela.problem-packet.v0.1","problem":30,"statement":"Let $h(N)$ be the maximum size of a Sidon set in $\\{1,\\ldots,N\\}$. Is it true that, for every $\\epsilon>0$,\\[h(N) = N^{1/2}+O_\\epsilon(N^\\epsilon)?\\]","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_ce606f2e709ba1a8","kind":"dead_end","claim":"attempted via frontier 'sidon/B2' (transfer_strength=partial) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_d6f9c531894b12a7","kind":"verified_witness","claim":"OEIS A309370 (max Sidon subset of {0,1}^n) a(23) >= 5179: an independently verified Sidon set of 5179 points in {0,1}^23 (all 13413610 pairwise componentwise sums distinct), extending the sequence to n=23 where no value was recorded. Built by lifting the best n=22 set (3770) into {0,1}^23 and greedily adding new-coordinate vectors; re-checked from scratch by an independent pairwise-sum-distinctness verifier.","grade":"new_oeis_term","gateStatus":"verified","superseded":false},{"id":"att_5e83a4dbbabefcdf","kind":"verified_witness","claim":"OEIS A309370 (max Sidon subset of {0,1}^n) a(24) >= 7179: an independently verified Sidon set of 7179 points in {0,1}^24 (all 25,772,610 pairwise componentwise sums distinct), extending the sequence to n=24 where no value was recorded. Built by lifting the verified n=23 set (5179) into {0,1}^24 and greedily adding new-coordinate vectors; re-checked from scratch by an independent pairwise-sum-distinctness verifier and by the public Blair-Link verify.py.","grade":"new_oeis_term","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[{"id":"A003022","name":"Length of shortest (or optimal) Golomb ruler with n marks.","terms":"1,3,6,11,17,25,34,44,55,72,85,106,127,151,177,199,216,246,283,333,356,372,425,480,492,553,585","url":"https://oeis.org/A003022"},{"id":"A143824","name":"Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.","terms":"0,1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10","url":"https://oeis.org/A143824"},{"id":"A227590","name":"a(n) = A003022(n)+1 with a(1)=1.","terms":"1,2,4,7,12,18,26,35,45,56,73,86,107,128,152,178,200,217,247,284,334,357,373,426,481,493,554,586","url":"https://oeis.org/A227590"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}