# Lower Bounds of Size Ramsey Number for Graphs with Small Independence Number

Lower Bounds of Size Ramsey Number for Graphs with Small Independence Number Let r ≥ 3 be an integer such that r − 2 is a prime power and let H be a connected graph on n vertices with average degree at least d and α(H) ≤ βn, where 0 < β < 1 is a constant. We prove that the size Ramsey number \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat R(H;r) > {{nd} \over 2}{(r - 2)^2} - C\sqrt n$$\end{document} for all sufficiently large n, where C is a constant depending only on r, d and β. In particular, for integers k ≥ 1, and r ≥ 3 such that r − 2 is a prime power, we have that there exists a constant C depending only on r and k such that \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat R(P_n^k;r) > kn{(r - 2)^2} - C\sqrt n - {{({k^2} + k)} \over 2}{(r - 2)^2}$$\end{document} for all sufficiently large n, where Pnk is the kth power of Pn. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Lower Bounds of Size Ramsey Number for Graphs with Small Independence Number

, Volume 37 (4) – Oct 1, 2021
6 pages

/lp/springer-journals/lower-bounds-of-size-ramsey-number-for-graphs-with-small-independence-7KcHq5C8Kg
Publisher
Springer Journals
Copyright © The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2021
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-021-1047-3
Publisher site
See Article on Publisher Site

### Abstract

Let r ≥ 3 be an integer such that r − 2 is a prime power and let H be a connected graph on n vertices with average degree at least d and α(H) ≤ βn, where 0 < β < 1 is a constant. We prove that the size Ramsey number \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat R(H;r) > {{nd} \over 2}{(r - 2)^2} - C\sqrt n$$\end{document} for all sufficiently large n, where C is a constant depending only on r, d and β. In particular, for integers k ≥ 1, and r ≥ 3 such that r − 2 is a prime power, we have that there exists a constant C depending only on r and k such that \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat R(P_n^k;r) > kn{(r - 2)^2} - C\sqrt n - {{({k^2} + k)} \over 2}{(r - 2)^2}$$\end{document} for all sufficiently large n, where Pnk is the kth power of Pn.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Oct 1, 2021

Keywords: size ramsey number; affine plane; probabilistic method; 05C55; 05D10