Variants of Baranyai's Theorem with Additional Conditions

TL;DR

This paper extends Baranyai's theorem by constructing large families of partial partitions with controlled overlaps, and introduces cyclic orderings to generate consecutive parpartitions that are sufficiently distinct.

Contribution

It provides new bounds for the existence of non-overlapping parpartitions and introduces cyclic orderings to produce consecutive parpartitions with minimal overlap.

Findings

Constructed large families of $(k, ext{ell})$-parpartitions with no two being $( ext{alpha}, ext{beta})$-close.

Improved conditions over previous results by Katona and Katona for the existence of such partitions.

Established cyclic orderings that generate consecutive parpartitions with controlled overlap properties.

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.