Integer sequences without singleton intersection

Let $𝒜 (n, k)$ denote the collection of all $n^{k}$ integer sequences $(a_{1}, a_{2}, \dots, a_{k})$ , $1 \leq a_{i} \leq n$ . Let $m (n, k)$ be the maximum of $| ℱ |$ , $ℱ \subset 𝒜 (n, k)$ and no two sequences in $ℱ$ coincide in exactly one coordinate. The main results are $m (n, k) = n^{k - 2}$ for $k \geq 6$ and $n \geq n_{0} (k)$ , $m (n, 5) = (1 + o (1)) n^{3}$ and $m (n, 4) = 1 + 2 {(n - 1)}^{2}$ for $n \geq 3$ . The integer sequence version of the Deza–Erdős–Frankl theorem is established as well.

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Keywords

integer sequences, $L$-intersecting, singleton intersection

Mathematical Subject Classification

Primary: 05D05

Milestones

Received: 6 November 2022

Revised: 18 January 2023

Accepted: 2 March 2023

Published: 4 June 2023

Authors

Péter Frankl
Rényi Institute Hungarian Academy of Sciences Budapest Hungary

Jian Wang
Department of Mathematics Taiyuan University of Technology Taiyuan China

Vol. 12, No. 2, 2023

Péter Frankl and Jian Wang

Abstract

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Keywords

Mathematical Subject Classification

Milestones

Authors