La questione è decisamente molto più seria di quello che pensavo
Ho provato a fare qualche ricerca anch'io, non ho trovato la dimostrazione per il pentagono ma quella per l'esagono sì
A questo
link c'è una soluzione per l'esagono convesso, trovata al computer però.
Per quanto riguarda i pentagoni, nel documento si afferma questo:
Testo nascosto, fai click qui per vederlo
"... At the time when the conjectures (Pk) and (Qk) were proposed they seemed to be a hazardous extrapolation from a few trivial and not very convincing cases, but a few years later Makai and Turan verified that indeed 9 points always contain a convex pentagon. As far as we know this proof has never appeared in print but some of us who knew of its existence saw in it modest support for the belief that (Pk) is true for all k. (See Morris and Soltan [7], Kalbfleisch etal. [5] and Bonnice [1] for proofs for 9 points.) Many years later Erdos and Szekeres [4] produced an explicit example of 2k~2 points which contained no convex k-gon, thereby confirming the truth of (Qk) for all k > 1. This construction showed at any rate that no(k) > 2k~2. For recent results on the upper bound for no(k) see Toth and Valtr [8]. ..."
Cordialmente, Alex