## BiPella: Mini Review on a Novel Method to Treat Cardiogenic Shock Patients

ACTA CARDIOLOGICA SINICA, vol.37, no.3, pp.312-317, 2021 (Peer-Reviewed Journal)

• Publication Type: Article / Letter
• Volume: 37 Issue: 3
• Publication Date: 2021
• Doi Number: 10.6515/acs.202105_37(3).20210118c
• Journal Name: ACTA CARDIOLOGICA SINICA
• Journal Indexes: Science Citation Index Expanded, Scopus, CAB Abstracts, EMBASE, Veterinary Science Database
• Page Numbers: pp.312-317

#### Abstract

The set of the graphs which do not contain the complete graph on $q$ vertices $K_q$ and have the property that in every coloring of their edges in two colors there exist a monochromatic triangle is denoted by $\mathcal{H}_e(3, 3; q)$. The edge Folkman numbers $F_e(3, 3; q) = \min\{|V(G)| : G \in \mathcal{H}_e(3, 3; q)\}$ are considered. Folkman proved in 1970 that $F_e(3, 3; q)$ exists if and only if $q \geq 4$. From the Ramsey number $R(3, 3) = 6$ it becomes clear that $F_e(3, 3; q) = 6$ if $q \geq 7$. It is also known that $F_e(3, 3; 6) = 8$ and $F_e(3, 3; 5) = 15$. The upper bounds on the number $F_e(3, 3; 4)$ which follow from the construction of Folkman and from the constructions of some other authors are not good. In 1975 Erdo\"s posed the problem to prove the inequality $F_e(3, 3; 4) < 10^{10}$. This Erdo\"s problem was solved by Spencer in 1978. The last upper bound on $F_e(3, 3; 4)$ was obtained in 2012 by Lange, Radziszowski and Xu, who proved that $F_e(3, 3; 4) \leq 786$. The best lower bound on this number is 19 and was obtained 10 years ago by Radziszowski and Xu. In this paper, we improve this result by proving $F_e(3, 3; 4) \geq 20$. At the end of the paper, we improve the known bounds on the vertex Folkman number $F_v(2, 3, 3; 4)$ by proving $20 \leq F_v(2, 3, 3; 4) \leq 24$.