In this paper, we will define a function $H:\N \rightarrow \N$, whose output is the size of an optimized hypergraph based upon the restraints given by its input values. This function is known to be well-defined, however its values are unknown for larger $n\in \N$. Only an upper and lower bound for this function are definitively known. Here, we will use properties of pre-ordered sets to define an improved lower bound for $H$.
