Involve: A Journal of Mathematics

  • Involve
  • Volume 8, Number 1 (2015), 119-127.

An Erdős–Ko–Rado theorem for subset partitions

Adam Dyck and Karen Meagher

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/involve.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A k-subset partition, or (k,)-subpartition, is a k-subset of an n-set that is partitioned into distinct blocks, each of size k. Two (k,)-subpartitions are said to t-intersect if they have at least t blocks in common. In this paper, we prove an Erdős–Ko–Rado theorem for intersecting families of (k,)-subpartitions. We show that for n k, 2 and k 3, the number of (k,)-subpartitions in the largest 1-intersecting family is at most nk k n2k k n(1)k k ( 1)!, and that this bound is only attained by the family of (k,)-subpartitions with a common fixed block, known as the canonical intersecting family of (k,)-subpartitions. Further, provided that n is sufficiently large relative to k, and t, the largest t-intersecting family is the family of (k,)-subpartitions that contain a common set of t fixed blocks.

Article information

Source
Involve, Volume 8, Number 1 (2015), 119-127.

Dates
Received: 3 October 2013
Revised: 9 April 2014
Accepted: 12 April 2014
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511370844

Digital Object Identifier
doi:10.2140/involve.2015.8.119

Mathematical Reviews number (MathSciNet)
MR3321715

Zentralblatt MATH identifier
1309.05176

Subjects
Primary: 05D05: Extremal set theory

Keywords
Erdős–Ko–Rado theorem set partitions

Citation

Dyck, Adam; Meagher, Karen. An Erdős–Ko–Rado theorem for subset partitions. Involve 8 (2015), no. 1, 119--127. doi:10.2140/involve.2015.8.119. https://projecteuclid.org/euclid.involve/1511370844


Export citation

References

  • R. Ahlswede and L. H. Khachatrian, “The complete intersection theorem for systems of finite sets”, European J. Combin. 18:2 (1997), 125–136.
  • F. Brunk and S. Huczynska, “Some Erdős–Ko–Rado theorems for injections”, European J. Combin. 31:3 (2010), 839–860.
  • P. J. Cameron and C. Y. Ku, “Intersecting families of permutations”, European J. Combin. 24:7 (2003), 881–890.
  • P. Erdős, “My joint work with Richard Rado”, pp. 53–80 in Surveys in combinatorics 1987 (New Cross, 1987), edited by C. Whitehead, London Math. Soc. Lecture Note Ser. 123, Cambridge Univ. Press, 1987.
  • P. Erdős, C. Ko, and R. Rado, “Intersection theorems for systems of finite sets”, Quart. J. Math. Oxford Ser. $(2)$ 12 (1961), 313–320.
  • P. Frankl and R. M. Wilson, “The Erdős–Ko–Rado theorem for vector spaces”, J. Combin. Theory Ser. A 43:2 (1986), 228–236.
  • V. Kamat and N. Misra, “An Erdős–Ko–Rado theorem for matchings in the complete graph”, preprint, 2013.
  • C. Y. Ku and I. Leader, “An Erdős–Ko–Rado theorem for partial permutations”, Discrete Math. 306:1 (2006), 74–86.
  • C. Y. Ku and D. Renshaw, “Erdős–Ko–Rado theorems for permutations and set partitions”, J. Combin. Theory Ser. A 115:6 (2008), 1008–1020.
  • K. Meagher and L. Moura, “Erdős–Ko–Rado theorems for uniform set-partition systems”, Electron. J. Combin. 12 (2005), Research Paper 40.
  • B. M. I. Rands, “An extension of the Erdős, Ko, Rado theorem to $t$-designs”, J. Combin. Theory Ser. A 32:3 (1982), 391–395.
  • R. M. Wilson, “The exact bound in the Erdős–Ko–Rado theorem”, Combinatorica 4:2-3 (1984), 247–257.