Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors
- 1Department of Mathematics, University of North Dakota, Grand Forks, ND 58202, USA
Abstract
The Ramsey number R(s,t) is the least integer n such that any coloring of the edges of Kn with two colors produces either a monochromatic Ks in one color or a monochromatic Kt in the other. If s=t, we call R(s,s) a diagonal Ramsey number. The threshold Ramsey multiplicity of a diagonal Ramsey number R(s,s), denoted by m(s,s) or m2(s), is the smallest number of copies of a monochromatic Ks that can be found in any coloring of the edges of KR(s,s). For instance, m2(2)=1, m2(3)=2, and m2(4)=9. We derive upper bounds for multicolor, off-diagonal threshold Ramsey multiplicities. In the diagonal two-color case, the resulting explicit numerical bounds improve those obtained from the elementary random-coloring estimate for 5≤s≤8. In the multicolor case, we recover the known value m(3,3,3)=5 and obtain the bound m(3,3,4)≤8. We conclude with a general framework for seeking further improvements.
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BibTeX
@article{christopherson2026bounds,
title = {{Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors}},
author = {Bryce Alan Christopherson and Casia Steinhaus},
journal = {Mathematics},
year = {2026},
month = sep,
doi = {10.3390/math14193553},
url = {https://www.mdpi.com/2227-7390/14/19/3553},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors
AU - Bryce Alan Christopherson
AU - Casia Steinhaus
PY - 2026
DA - 2026/09/30
JO - Mathematics
DO - 10.3390/math14193553
UR - https://www.mdpi.com/2227-7390/14/19/3553
AB - The Ramsey number R(s,t) is the least integer n such that any coloring of the edges of Kn with two colors produces either a monochromatic Ks in one color or a monochromatic Kt in the other. If s=t, we call R(s,s) a diagonal Ramsey number. The threshold Ramsey multiplicity of a diagonal Ramsey number R(s,s), denoted by m(s,s) or m2(s), is the smallest number of copies of a monochromatic Ks that can be found in any coloring of the edges of KR(s,s). For instance, m2(2)=1, m2(3)=2, and m2(4)=9. We derive upper bounds for multicolor, off-diagonal threshold Ramsey multiplicities. In the diagonal two-color case, the resulting explicit numerical bounds improve those obtained from the elementary random-coloring estimate for 5≤s≤8. In the multicolor case, we recover the known value m(3,3,3)=5 and obtain the bound m(3,3,4)≤8. We conclude with a general framework for seeking further improvements.
SN - 2227-7390
ER -