The critical number $cr(r,n)$ of natural intervals $[r,n]$ was introduced by Herzog, Kaplan and Lev in 2014. The critical number $cr(r,n)$ is the smallest integer $t$ satisfying the following conditions: (i) every sequence of integers $S=\{r_1=r\leq r_2\leq \dotsb\leq r_k\}$ with $r_1+r_2 +\dotsb +r_k=n$ and $k\geq t$ has the following property: every integer between $r$ and $n-r$ can be written as a sum of distinct elements of $S$, and (ii) there exists $S$ with $k=t$, which satisfies that property. In this paper we study a variation of the critical number $cr(r,n)$ called the $r$-critical number $rcr(r,n)$. We determine the value of $rcr(r,n)$ for all $r,n$ satisfying $r\mid n$.
@article{10_37236_9835,
author = {Marcel Herzog and Gil Kaplan and Arieh Lev and Romina Zigdon},
title = {\(r\)-critical numbers of natural intervals},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/9835},
zbl = {1511.11005},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9835/}
}
TY - JOUR
AU - Marcel Herzog
AU - Gil Kaplan
AU - Arieh Lev
AU - Romina Zigdon
TI - \(r\)-critical numbers of natural intervals
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/9835/
DO - 10.37236/9835
ID - 10_37236_9835
ER -
%0 Journal Article
%A Marcel Herzog
%A Gil Kaplan
%A Arieh Lev
%A Romina Zigdon
%T \(r\)-critical numbers of natural intervals
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/9835/
%R 10.37236/9835
%F 10_37236_9835
Marcel Herzog; Gil Kaplan; Arieh Lev; Romina Zigdon. \(r\)-critical numbers of natural intervals. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/9835
