Cycle-complete ramsey numbers

The Ramsey number r(Cℓ, Kn) is the smallest natural number N such that every red/blue edge-colouring of a clique of order N contains a red cycle of length ℓ or a blue clique of order n. In 1978, Erdos, Faudree, Rousseau and Schelp conjectured that r(Cℓ, Kn) = (ℓ − 1)(n − 1) + 1 for ℓ ≥ n ≥ 3 provided (ℓ, n) 6= (3, 3). We prove that, for some absolute constant C ≥ 1, we have r(Cℓ, Kn) = (ℓ − 1)(n − 1) + 1 provided ℓ ≥ C logloglognn. Up to the value of C this is tight since we also show that, for any ε > 0 and n > n0(ε), we have r(Cℓ, Kn) ≫ (ℓ − 1)(n − 1) + 1 for all 3 ≤ ℓ ≤ (1 − ε)logloglognn. This proves the conjecture of Erdos, Faudree, Rousseau and Schelp for large ℓ, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erdos, Faudree, Rousseau and Schelp

Cycle-complete Ramsey numbers

The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erd\H{o}s, Faudree, Rousseau and Schelp conjectured that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ provided $\ell \geq C\frac {\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon >0$ and $n> n_0(\varepsilon )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 \leq \ell \leq (1-\varepsilon )\frac {\log n}{\log \log n}$. This proves the conjecture of Erd\H{o}s, Faudree, Rousseau and Schelp for large $\ell$, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erd\H{o}s, Faudree, Rousseau and Schelp.Comment: 19 page

R(W5 , K5) = 27

The two-color Ramsey number R(G , H) is defined to be the smallest integer n such that any graph F on n vertices contains either a subgraph isomorphic to G or the complement of F contains a subgraph isomorphic to H. Ramsey numbers serve to quantify many of the existing theorems of Ramsey theory, which looks at large combinatorial objects for certain given smaller combinatorial objects that must be present. In 1989 George R. T. Hendry presented a table of two-color Ramsey numbers R(G , H) for all pairs of graphs G and H having at most five vertices. This table left seven unsolved cases, of which three have since been solved. This thesis eliminates one of the remaining four cases, R(W5 , K5), where a K5 is the complete graph on five vertices and a W5 is a wheel of order 5, which can be pictured as a wheel having four spokes or as a cycle of length 4 having all four vertices adjacent to a central vertex. In this thesis we show R(W5, K5) to be equal to 27, utilizing a combinatorial approach with significant computations. Specifically we use a technique developed by McKay and Radziszowski to effectively glue together smaller graphs in an effort to prove exhaustively that no graph having 27 vertices exists that does not contain an independent set on five vertices or a subgraph isomorphic to W5. The previous best bounds for this case were 27 \u3c= R( W_5 , K_5 ) \u3c= 29