## Abstract

Let k be a positive integer. A sequence s_{1}, s_{2},.., s_{m} over an n-element A alphabet is a packing k-radius sequence, if for all pairs of indices (i, j), such that 1≤i<j≤m and j-i≤k, the sets {s_{i}, s_{j}} are pairwise different 2-element subsets of A. Let g_{k}(n) denote the length of a longest k-radius sequence over A. We give a construction demonstrating that for every k=⌊cn^{α}⌋, where c and α are fixed reals such that c>0 and 0≤α<1/2, g_{k}(n)=n^{2}/2k(1-o(1)). For a constant k we show that g_{k}(n)=n^{2}/2k-O(n^{1.525}). Moreover, we prove an upper bound for g_{k}(n) that allows us to show that g_{k}(n)=n(1+o(1)) for every k=⌊cn^{α}⌋, where c>0 and 1/2<α<1.

Original language | English |
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Pages (from-to) | 57-70 |

Number of pages | 14 |

Journal | European Journal of Combinatorics |

Volume | 57 |

DOIs | |

State | Published - Oct 1 2016 |

### Bibliographical note

Funding Information:Zbigniew Lonc acknowledges a support from the Polish National Science Centre , decision no. DEC-2012/05/B/ST1/00652 .

Publisher Copyright:

© 2016 Elsevier Ltd.

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics