Restricted Size Ramsey Number for Path of Order Three Versus Graph of Order Five

Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for a pair of graph $G$ and $H$ is the smallest number $\hat{r}$ such that there exists a graph $F$ with size $\hat{r}$ satisfying the property that any red-blue coloring of edges of $F$ contains a red subgraph $G$ or a blue subgraph $H$. Additionally, if the order of $F$ in the size Ramsey number is $r(G,H)$, then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller\u27s works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five

Restricted Size Ramsey Number for Matching versus Tree and Triangle Unicyclic Graphs of Order Six

Let F, G, and H be simple graphs. The graph F arrows (G,H) if for any red-blue coloring on the edge of F, we find either a red-colored graph G or a blue-colored graph H in F. The Ramsey number r(G,H) is the smallest positive integer r such that a complete graph Kr arrows (G,H). The restricted size Ramsey number r∗(G,H) is the smallest positive integer r∗ such that there is a graph F, of order r(G,H) and with the size r∗, satisfying F arrows (G,H). In this paper we give the restricted size Ramsey number for a matching of two edges versus tree and triangle unicyclic graphs of order six

Ramsey numbers for partially-ordered sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl