{"schema":"vela.problem-packet.v0.1","problem":703,"statement":"Let $r\\geq 1$ and define $T(n,r)$ to be maximal such that there exists a family $\\mathcal{F}$ of subsets of $\\{1,\\ldots,n\\}$ of size $T(n,r)$ such that $\\lvert A\\cap B\\rvert\\neq r$ for all $A,B\\in \\mathcal{F}$.Estimate $T(n,r)$ for $r\\geq 2$. In particular, is it true that for every $\\epsilon>0$ there exists $\\delta>0$ such that for all $\\epsilon n<r<(1/2-\\epsilon) n$ we have\\[T(n,r)<(2-\\delta)^n?\\]","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A390645","name":"Triangle read by rows: T(n,r) is maximal such that there exists a family F of subsets of {1,...,n} of size T(n,r) such that the intersection of no two sets in F has r elements.","terms":"1,1,2,2,3,4,4,4,7,8,8,6,11,15,16,16,11,16,26,31,32,32,23,22,42,57,63,64,64,45,37,64,99,120,127,128,128,94,67,93,163,219,","url":"https://oeis.org/A390645"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}