Two Conjectures On Ramsey Numbers And Related Research

For two graphs G₁and G₂,the online Ramsey number（?）（G₁,G₂）is the smallest number of rounds（or equivalently,edges）that Builder draws on an infinite empty graph to guarantee that there is either a red copy of G₁or a blue copy of G₂,under the condition that Builder draws one edge in each round and Painter immediately colors it red or blue.In 2015,Cyman,Dzido,Lapinskas,and Lo conjectured that（?）(P₄,P₊₁)=（7+2）/5for all≥3.This conjecture remains open for eight years.When is small,there are only a few exact online Ramsey numbers of P₄versus P.The numbers of（?）（P₄,P）were calculated with the help of computers for 6≤≤9.In 2021,Dzido and Zakrzewska verified that（?）(P₄,P₁₀)=13 and（?）(P₄,P₁₁)=15.In the second chapter of this thesis,we solve the conjecture by creating Builder’s strategy of choosing edges.For two graphs G₁and G₂,the connected size Ramsey number（?）_c（G₁,G₂）is the smallest number of edges of a connected graph G such that for any partition（E₁,E₂）of E（G）,either G₁?E₁or G₂?E₂.Let n K₂be a matching with n edges.Rahad-jeng,Baskoro,and Assiyatun conjectured that（?）_c（n K₂,P₄）=3n-1 if n is even,and r?_c（n K₂,P₄）=3n otherwise.In Chapter 3,we completely solve the conjecture by intro-ducing the concept of deletable edge set and analyzing the end blocks carefully.The online Ramsey numbers involving a star K_1,nalso attracts some researchers.In the fourth chapter of this thesis,we give the exact value of（?）(K_1,n,P₄).We define B_nto be the graph formed by n triangles sharing a common edge.We initiate the study of online Ramsey numbers involving B_nand determine the exact value of（?）(K_1,2,B_n)in Chapter 4.