Cibulka, Kyncl and Valtr have solved problem 3. They constructed point sets with no empty pentagon, and whose visibility graph has arbitrarily large clique number.
Josef Cibulka, Jan Kyncl and Pavel Valtr: On planar point sets with the pentagon property, Proc. SoCG 2013, pp. 81-90. http://dl.acm.org/citation.cfm?id=2462406
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Problem 3 solved
Cibulka, Kyncl and Valtr have solved problem 3. They constructed point sets with no empty pentagon, and whose visibility graph has arbitrarily large clique number.
Josef Cibulka, Jan Kyncl and Pavel Valtr: On planar point sets with the pentagon property, Proc. SoCG 2013, pp. 81-90. http://dl.acm.org/citation.cfm?id=2462406