An $8+9$ gluing computation for the Ramsey number $R(F_3,F_4)$

Source abstract

Let $F_t$ be the graph consisting of $t$ triangles with a common vertex and otherwise disjoint vertices. We prove that $R(F_3,F_4)=17$ using an $8+9$ gluing computation. Neighbourhood matching arguments give degree and clique restrictions that reduce the search on seventeen vertices to $88,294,674$ ordered seed pairs. The verified main computation excludes $88,294,634$ pairs. A cross-edge counting argument excludes thirty-four of the forty exceptional pairs, and independently checked finite-domain certificates exclude the remaining six. Together with the construction whose red graph is the complete bipartite graph $K_{8,8}$, these results establish $R(F_3,F_4)=17$.