Study Of Two Types Of Ramsey Numbers

In 1930,British mathematician Frank Ramsey published the seminal paper On a Problem of Formal Logic,which is regarded as the beginning of Ramsey theory.For given graphs G₁,...,G_k,r（G₁,...,G_k）is the minimum integer n that satisfies the following condition:any k-edge-coloring of K_ncontains a copy of G_iin color i for some i with1≤i≤k.After many mathematicians’continuous research and development,the theory is finally formed.The core of Ramsey theory is that complete disorder is impossible.The online Ramsey number and the connected size Ramsey number are important variations of the original Ramsey number.There are few research results on the two types of Ramsey numbers for sparse graphs.Most of the literatures only contain some upper and lower bounds.Beck creatively introduced the online Ramsey game and started a new research field,which is not only a meaningful development of Ramsey theory,but also a new topic to be explored in graph theory itself.In 2020,Dzido et al.proved that（?）（C₃,P_l）≤4l-5.In Chapter 2,we show that（?）（C₃,P_l）≤3l-3.In the same chapter,we also obtain new upper bounds of（?）（C₄,P_l）and（?）（C₄,C_l）.At the same time,we deduce that（?）（P₃,W_l）=（?）9l/4（?）+1.Erd?os et al.proposed the oncept of size Ramsey number.The concept of connected size Ramsey number is derived from the development of size Ramsey number,which greatly enriches graph Ramsey theory.In 2017,Rahadjeng et al.proved that（?）_c（n K₂,C₄）≤5n-1.In Chapter 3,we improve the previous result by showing（?）_c（n K₂,C₄）≤（?）（9n-1）/2（?）.Also,we prove that（?）_c（n K₂,P₃）=（?）（5n-1）/2（?）,（?）_c（n K₂,2P₃）=（?）（5n+9）/2（?）,and （?）_c（n K₂,C₃）=4n-1.