- Series
- Graph Theory Seminar
- Time
- Tuesday, April 26, 2022 - 11:00am for 1 hour (actually 50 minutes)
- Location
- Skiles 005/Zoom (hybrid)
- Speaker
- Meysam Miralaei – Institute for Research in Fundamental Sciences, Iran – m.miralaei@ipm.ir – https://scholar.google.com/citations?user=jnEe2D8AAAAJ&hl=en
- Organizer
- Anton Bernshteyn

**Please Note:** Note the unusual time!

For given graphs $G$ and $H$ and a graph $F$, we say that $F$ is Ramsey for $(G, H)$ and we write $F \longrightarrow (G,H)$, if for every $2$-edge coloring of $F$, with colors red and blue, the graph $F$ contains either a red copy of $G$ or a blue copy of $H$. A natural question is how few vertices can a graph $F$ have, such that $F \longrightarrow (G,H)$? Frank P. Ramsey studied this question and proved that for given graphs $G$ and $H$, there exists a positive integer $n$ such that for the complete graph $K_n$ we have $ K_n \longrightarrow (G,H)$. The smallest such $n$ is known as the Ramsey number of $G$, $H$ and is denoted by $R(G, H)$. Instead of minimizing the number of vertices, one can ask for the minimum number of edges of such a graph, i.e. can we find a graph which possibly has more vertices than $R(G, H)$, but has fewer edges and still is Ramsey for $(G,H)$? How many edges suffice to construct a graph which is Ramsey for $(G,H)$? The attempts at answering the last question give rise to the notion of size-Ramsey number of graphs. In 1978, Erdős, Faudree, Rousseau and Schelp pioneered the study of the size-Ramsey number to be the minimum number of edges in a graph $F$, such that $F$ is Ramsey for $(G,H)$. In this talk, first I will give a short history about the size Ramsey number of graphs with a special focus on sparse graphs. Moreover, I will talk about the multicolor case of the size Ramsey number of cycles with more details.