Erdős and Szekeres proved the bounds
$$
2^{n-2} + 1 ≤ f(n) ≤ \binom{2n-4}{n-2} + 1
$$
([ErSz60] and [ErSz35] respectively).
[ErSz60] Erdős, P. and Szekeres, G., _On some extremum problems in elementary geometry_.
Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1960/61), 53-62.
[ErSz35] Erdős, P. and Szekeres, G., _A combinatorial problem in geometry_.
Compos. Math. (1935), 463-470.
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