(a) If S ⊂ Z+ and |S| >3, prove that there exist distinct x, y e S where x + y is even.
(b) Let S ⊂ Z+ × Z+. Find the minimal value of |S| that guarantees the existence of distinct ordered pairs (x1, x2), (y1, y2) ∈ S such that x1 + y1 and x2 + y2 are both even.
(c) Extending the ideas in parts (a) and (b), consider S ∈ Z+ × Z+ × Z+. What size must |5| be to guarantee the existence of distinct ordered triples (x1, x2, x3), (y1, y2, y3) ∈ S where x1 + y1, x2 + y2 and pj + yj are all even?
d) Generalize the results of parts (a), (b), and (c).
e) A point P(x, y) in the Cartesian plane is called a lattice point if x, y ∈ Z. Given distinct lattice points P1(x1 y1), P2(x2, y2),, Pn(xn, yn), determine the smallest value of n that guarantees the existence of Pi(xt, yi), Pj(Xj, yj), 1 < i < j < n, such that the midpoint of the line segment connecting Pi(xt, yt) and Pj (xj, yj) is also a lattice point.