Critical numbers of intervals

TextIn 2014, Herzog, Kaplan and Lev introduced the critical numbers cr(r,n)cr(r,n) of natural intervals [r,n][r,n] and determined the values of cr(1,n)cr(1,n) and cr(2,n)cr(2,n) for all n  , and for r≥3r≥3, they determined the values of cr(r,n)cr(r,n) for n≤(3r2+3r)/2n≤(3r2+3r)/2 and n≥(5r2+r)/2n≥(5r2+r)/2. In this note, we determine the remaining values of cr(r,n)cr(r,n) for r≥3r≥3.VideoFor a video summary of this paper, please visit https://youtu.be/hZSI-Ul3K94.