### 26th Putnam 1965

Problem B5

Let S be a set with n > 3 elements. Prove that we can find a collection of [n2/4] 2-subsets of S such that for any three distinct elements A, B, C of the collection, A ∪ B ∪ C has at least 4 elements.

Solution

This is just another form of finding a graph without triangles. If n = 2m, then take S to be the disjoint union of U and V, where U and V each have m elements. Now take K to be the collection all sets {u, v} with u in U and v in V. Evidently K has m2 = [n2/4] members. Suppose we have three distinct elements A, B, C of K. A ∪ B ∪ C cannot have less than three elements, for then two of A, B, C would be identical. So if it has less than 4 elements, then it has just three elements. But that means that for some a, b, c, we have A = {b, c}, B = {a, c}, C = {a, b}. Suppose a is in U. Then, considering B and C, c and b must be in V, but then A has two elements in V. Contradiction. Similarly if a is in V. So K has the required property.

Similarly, if n = 2m+1, take S to be the disjoint union of U and V, where U has m+1 elements and V has m elements. Then K, the set of all {u, v} with u in U and v in V has m(m+1) = [n2/4] elements and has the required property by the same argument.