{"schema":"vela.problem-packet.v0.1","problem":14,"statement":"Let $A\\subseteq \\mathbb{N}$. Let $B\\subseteq \\mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.Is it true that for all $\\epsilon>0$ and large $N$\\[\\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}?\\]Is it possible that\\[\\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert =o(N^{1/2})?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_17263d48ce0ed0b9","kind":"dead_end","claim":"attempted via frontier 'sidon/B2' (transfer_strength=weak) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[{"id":"A143824","name":"Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences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"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}