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Variants of Baranyai's Theorem with Additional Conditions

Zoe Xi

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 $(α,β)$-close for $α,β\in (0,1)$ if there exist subsets $A_1\neq B_1\in P_1$ and $A_2\neq B_2\in P_2$ such that $|A_1\cap A_2| > α{k}$ and $|B_1\cap B_2| > β{k}$. We establish that, given integers $k$, $\ell$, and $n$ satisfying $k^2\ell\leq n$ and $α, β\in (0, 1)$ satisfying $α+β\geq{(k+2)/k}$, one can find $\lfloor {n\choose k}/\ell\rfloor$ $(k, \ell)$-parpartitions of $[n]$ such that no two distinct $(k, \ell)$-parpartitions are $(α,β)$-close; this result improves the condition $k=O(1)$ and $\ell=o(\sqrt{n})$ in a corresponding result by Katona and Katona for $α= β= 1/2$. 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 $α+β\geq{1}$ 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 $(α,β)$-close.

Variants of Baranyai's Theorem with Additional Conditions

Abstract

A classical theorem of Baranyai states that, given integers such that divides , one can find a family of partitions of into -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 that consist of -element sets not overlapping significantly. More precisely, two parpartitions and are considered -close for if there exist subsets and such that and . We establish that, given integers , , and satisfying and satisfying , one can find -parpartitions of such that no two distinct -parpartitions are -close; this result improves the condition and in a corresponding result by Katona and Katona for . We also prove that, given integers , , and satisfying and , there is a cyclic ordering of the -element subsets of for any chosen such that any consecutive -element subsets in the ordering form a -parpartition of , which we refer to as a consecutive -parpartition (according to the ordering), and any two of these disjoint consecutive -parpartitions are not -close.

Paper Structure

This paper contains 8 sections, 12 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our results.
  3. Outline of the paper.
  4. Preliminaries
  5. Existence of linear-sized "not-too-close" $(k, \ell)$-parpartitions of $[n]$
  6. Existence of $(\ell, 2, \ell, 2)$-bag-avoiding powers of Hamiltonian cycles
  7. Conclusion
  8. Acknowledgements

Key Result

Theorem 1.1

Let $2\leq k < n$ be integers such that $k$ divides $n$. There exist ${n-1\choose k-1}$ partitions of $[n]$ into $k$-element subsets such that no subset appears in two of these partitions.

Figures (1)

  • Figure 1: A drawing of some $H^2$ with one edge, colored red (dashed), missing. The rest of the edges in $H^2$ are solid lines, colored either blue or black. The cycle in blue (consisting of both dotted and solid lines) is a new Hamiltonian cycle $H'$ such that $H'^2$ is (fully) contained in the graph $G$; to show this, we consider a copy of $H[4]$ surrounding the missing edge (here consisting of the vertices $v_{11}, v_{12}, v_1, v_2$) and another disjoint copy of $H[4]$ (here consisting of the vertices $v_6, v_7, v_8, v_9$), which together form $K_{4, 4}$. (The existence of the other disjoint copy of $H[4]$ is given by Claim \ref{['claim:cor-bipartite']}.) For instance, the two dotted blue lines refer to two edges in this $K_{4,4}$. Essentially, because of this $K_{4, 4}$, in the $H'^2$ induced by the new ordering $H'$, the vertices that have new neighbors (that is, neighbors different from theirs in $H^2$) are indeed connected by an edge to each of these new neighbors.

Theorems & Definitions (27)

  • Theorem 1.1: Baranyai 1973
  • Theorem 1.2
  • Theorem 1.3: Theorem 2.1, Katona and Katona 2023
  • Theorem 1.4
  • Theorem 1.5: Dirac 1952
  • Theorem 1.6: Komlós, Sárközy, and Szemerédi 1998
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • ...and 17 more