Variants of Baranyai's Theorem with Additional Conditions

Abstract

A classical theorem of Baranyai states that, given integers $2\leq k < n$ such that $k$ divides $n$, one can find a family of ${n-1\choose k-1}$ partitions of $[n]$ into $k$-element subsets such that every subset appears in exactly one partition. In this paper, we build on recent work by Katona and Katona in studying partial partitions, or parpartitions, of $[n]$ that consist of $k$-element sets not overlapping significantly. More precisely, two parpartitions $P_1$ and $P_2$ are considered $(\alpha,\beta)$-close for $\alpha,\beta\in (0,1)$ if there exist subsets $A_1\neq B_1\in P_1$ and $A_2\neq B_2\in P_2$ such that $|A_1\cap A_2| > \alpha{k}$ and $|B_1\cap B_2| > \beta{k}$. We establish that, given integers $k$, $\ell$, and $n$ satisfying $k^2\ell\leq n$ and $\alpha, \beta\in (0, 1)$ satisfying $\alpha+\beta\geq{(k+2)/k}$, one can find $\lfloor {n\choose k}/\ell\rfloor$ $(k, \ell)$-parpartitions of $[n]$ such that no two distinct $(k, \ell)$-parpartitions are $(\alpha,\beta)$-close; this result improves the condition $k=O(1)$ and $\ell=o(\sqrt{n})$ in a corresponding result by Katona and Katona for $\alpha = \beta = 1/2$. We also prove that, given integers $k$, $\ell$, and $n$ satisfying $k=O(1)$ and $\ell=o(\sqrt{n})$, there is a cyclic ordering of the $k$-element subsets of $[n]$ for any chosen $\alpha+\beta\geq{1}$ such that any $\ell$ consecutive $k$-element subsets in the ordering form a $(k, \ell)$-parpartition of $[n]$, which we refer to as a consecutive $(k, \ell)$-parpartition (according to the ordering), and any two of these disjoint consecutive $(k, \ell)$-parpartitions are not $(\alpha,\beta)$-close.