An inductive approach to constructing Universal Cycles on the k-subsets of [n]

TL;DR

This paper presents a new inductive method for constructing Universal Cycles on k-subsets of [n], enabling the creation of such cycles for larger sets based on smaller known cycles.

Contribution

It introduces an inductive approach using sums and products of smaller cycles to construct Universal Cycles on larger sets, expanding the known existence results.

Findings

Universal Cycles constructed for specific cases

Method applies to sets where n ≥ 18 and n ≡ 2 mod 8

Provides a framework for future cycle constructions

Abstract

In this paper, we introduce a method of constructing Universal Cycles on sets by taking "sums" and "products" of smaller cycles. We demonstrate this new approach by proving that if there exist Universal Cycles on the 4-subsets of [18] and the 4-subsets of [26], then for any integer n which is greater than or equal 18 and equivalent to 2 mod 8, there exists a Universal Cycle on the 4-subsets of [n].