This is a theoretical note on a result in set theory. The focus of the discussion is on “large sets of t-designs.” Utilizing existing theorems, the author proves new theorems on the existence of large sets of prime sizes.

The first section introduces the work. The definitions of t-design and large sets of t-designs are provided. The second section is on construction methods for large sets of t-designs. This section begins with a collection of theorems from the literature on large sets and partitionable sets. Each theorem is of the nature, “If x exists, then y exists.” The x and y make a formula for the existence of large sets. These formulas are recursive and can be used for the construction of large sets. Later in the section, theorems on large sets of prime sizes, taken from the literature, are stated. The second section ends by proving new theorems on large sets. The third section uses the results in the second section and proves two new theorems on large sets of prime sizes. These theorems are revised for large sets of sizes two and three, for which more comprehensive results are obtained in the last section.

A background in set theory is required. The theorem statements and the proofs are written as concisely as possible.