Multicolor Ramsey number for triangles grows faster than for other odd cycles
Home
Search
Subjects
About Erdös
About This Site
Search
Subjects
 All (170)
 Ramsey Theory (40)
 Extremal Graph Theory (40)
 Coloring, Packing, and Covering (25)
 Random Graphs and Graph Enumeration (16)
 Hypergraphs (35)
 Infinite Graphs (14)
About Erdös
About This Site
For graphs \( G_i\), \( i=1,\dots,k\), let \( r(G_1,\dots,G_k)\) denote the smallest integer \( m\) satisfying the property that if the edges of the complete graph \( K_m\) are colored in \( k\) colors, then for some \( i\), \( 1\leq i\leq k\), there is a subgraph isomorphic to \( G_i\) with all edges in the \( i\)th color.
A multicolored Ramsey problem for odd cycles (proposed by Erdös and Graham [1])
Show that for \(n \geq 2\) and any \(k\), \[ \lim_{k \rightarrow \infty} \frac{r(\overbrace{C_{2n+1},\ldots,C_{2n+1}}^k)} { r(\underbrace{3,\ldots,3}_k)} = 0 \]This problem is open even for \( n=2\).
Bibliography  

1 
P. Erdös,
Some new problems and results in graph theory and
other branches of combinatorial mathematics, Combinatorics and
graph theory (Calcutta, 1980), Lecture Notes in Math., 885,
917, Springer, BerlinNew York, 1981.
