General position sets, colinear sets, and Sierpi\'nski product graphs

Let G f H denote the Sierpi\'nski product of G and H with respect to the function f. The Sierpi\'nski general position number gp ₒ (G, H) is introduced as the cardinality of a largest general position set in G f H over all possible functions f. Similarly, the lower Sierpi\'nski general position number gp ₒ (G, H) is the corresponding smallest cardinality. The concept of vertex-colinear sets is introduced. Bounds for the general position number in terms of extremal vertex-colinear sets, and bounds for the (lower) Sierpi\'nski general position number are proved. The extremal graphs are investigated. Formulas for the (lower) Sierpi\'nski general position number of the s with K₂ as the first factor are deduced. It is proved that if m, n 2, then gp ₒ (Kₘ, Kₙ) = m (n-1) and that if n 2m-2, then gp ₒ (Kₘ, Kₙ) = m (n-m+1).