{"schema":"vela.problem-packet.v0.1","problem":43,"statement":"If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that\\[ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1),\\]where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to\\[\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2}\\]for some constant $c>0$?","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"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)"}