Try this with a trapezium instead

Let $$S$$ be the set of lattice points $$(x,y)$$ with $$1 \leq x \leq 20$$ and $$1 \leq y \leq 30.$$ Let $$B \subseteq S$$ be a subset of $$S$$ such that no set of $$4$$ vertices in $$B$$ form a (non-degenerate) parallelogram. What is the largest possible number of elements in $$B?$$

Details and assumptions

A figure is degenerate if it has 0 area.

×