Some exact values on Ramsey numbers related to fans

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest integer $N$ such that any red-blue edge-coloring of the complete graph $K_N$ contains a red $F$ or a blue $H$. When $F=H$, we simply write $R_2(H)$. For an positive integer $n$, let $K_{1,n}$ be a star with $n+1$ vertices, $F_n$ be a fan with $2n+1$ vertices consisting of $n$ triangles sharing one common vertex, and $nK_3$ be a graph with $3n$ vertices obtained from the disjoint union of $n$ triangles. In 1975, Burr, Erd\H{o}s and Spencer \cite{B} proved that $R_2(nK_3)=5n$ for $n\ge2$. However, determining the exact value of $R_2(F_n)$ is notoriously difficult. So far, only $R_2(F_2)=9$ has been proved. Notice that both $F_n$ and $nK_3$ contain $n$ triangles and $|V(F_n)|<|V(nK_3)|$ for all $n\ge 2$. Chen, Yu and Zhao (2021) speculated that $R_2(F_n)\le R_2(nK_3)=5n$ for $n$ sufficiently large. In this paper, we first prove that $R(K_{1,n},F_n)=3n-\varepsilon$ for $n\ge1$, where $\varepsilon=0$ if $n$ is odd and $\varepsilon=1$ if $n$ is even. Applying the exact values of $R(K_{1,n},F_n)$, we will confirm $R_2(F_n)\le 5n$ for $n=3$ by showing that $R_2(F_3)=14$.Comment: 10 pages, 3 figure

Ramsey numbers r(K3, G) for connected graphs G of order seven

AbstractThe triangle-graph Ramsey numbers are determined for all 814 of the 853 connected graphs of order seven. For the remaining 39 graphs lower and upper bounds are improved

Ramsey numbers for sets of small graphs

AbstractThe Ramsey number r=r(G1-G2-⋯-Gm,H1-H2-⋯-Hn) denotes the smallest r such that every 2-coloring of the edges of the complete graph Kr contains a subgraph Gi with all edges of one color, of a subgraph Hi with all edges of a second color. These Ramsey numbers are determined for all sets of graphs with at most four vertices, and in the diagonal case (m=n,Gi=Hi) for all pairs of graphs, one with at most four and the other with five vertices, so as for all sets of graphs with five vertices

Mini-Workshop: Hypergraph Turan Problem

This mini-workshop focused on the hypergraph Turán problem. The interest in this difficult and old area was recently re-invigorated by many important developments such as the hypergraph regularity lemmas, flag algebras, and stability. The purpose of this meeting was to bring together experts in this field as well as promising young mathematicians to share expertise and initiate new collaborative projects