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Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed to find asymptotic results for very large graphs, so it seems that the method is not suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove the exact values $R(K_4^-,K_4^-,K_4^-)=28$, $R(K_8,C_5)= 29$, $R(K_9,C_6)= 41$, $R(Q_3,Q_3)=13$, $R(K_{3,5},K_{1,6})=17$, $R(C_3, C_5, C_5)= 17$, and $R(K_4^-,K_5^-;3)= 12$. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.
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In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,t$, we determine the complementary Ramsey numbers $bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.
A set of vertices $Xsubseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $xin X$ is either isolated in the induced subgraph $langle Xrangle$ or else has a private neighbor $yin Vsetminus X$ that is adjacent to $x$ and to no other vertex of $X$. The emph{irredundant Ramsey number} $s(m,n)$ is the smallest $N$ such that in every red-blue coloring of the edges of the complete graph of order $N$, either the blue subgraph contains an $m$-element irredundant set or the red subgraph contains an $n$-element irredundant set. The emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ yields an $m$-element irredundant set in the blue subgraph or an $n$-element independent set in the red subgraph. In this paper, we first improve the upper bound of $t(3,n)$; using this result, we confirm that a conjecture proposed by Chen, Hattingh, and Rousseau, that is, $lim_{nrightarrow infty}frac{t(m,n)}{r(m,n)}=0$ for each fixed $mgeq 3$, is true for $mleq 4$. At last, we prove that $s(3,9)$ and $t(3,9)$ are both equal to $26$.
We prove that the number of integers in the interval [0,x] that are non-trivial Ramsey numbers r(k,n) (3 <= k <= n) has order of magnitude (x ln x)**(1/2).
Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for $frac{8}{9}n+112le mle lceilfrac{3n}{2}rceil+1$ and $n geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, {it Ars Combin.,} {bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k geq 3$. Namely, we prove that for each $k geq 3$, $r(B_n^{(k)}, C_n)= (k+1+o_k(1))n.$ This extends a result due to Rousseau and Sheehan (A class of Ramsey problems involving trees, {it J.~London Math.~Soc.,} {bf 18} (1978), 392--396).
Burr and ErdH{o}s in 1975 conjectured, and Chvatal, Rodl, Szemeredi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr--ErdH{o}s conjecture, answering a question of Bucic, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented Ramsey number of $H$, denoted $overrightarrow{r_{1}}(H)$, is the least $N$ such that every tournament on $N$ vertices contains a copy of $H$. We show that for any $Delta geq 2$ and any sufficiently large $n$, there exists an acyclic digraph $H$ with $n$ vertices and maximum degree $Delta$ such that [ overrightarrow{r_{1}}(H)ge n^{Omega(Delta^{2/3}/ log^{5/3} Delta)}. ] This proves that $overrightarrow{r_{1}}(H)$ is not always linear in the number of vertices for bounded-degree $H$. On the other hand, we show that $overrightarrow{r_{1}}(H)$ is nearly linear in the number of vertices for typical bounded-degree acyclic digraphs $H$, and obtain linear or nearly linear bounds for several natural families of bounded-degree acyclic digraphs. For multiple colors, we prove a quasipolynomial upper bound $overrightarrow{r_{k}}(H)=2^{(log n)^{O_{k}(1)}}$ for all bounded-degree acyclic digraphs $H$ on $n$ vertices, where $overrightarrow{r_k}(H)$ is the least $N$ such that every $k$-edge-colored tournament on $N$ vertices contains a monochromatic copy of $H$. For $kgeq 2$ and $ngeq 4$, we exhibit an acyclic digraph $H$ with $n$ vertices and maximum degree $3$ such that $overrightarrow{r_{k}}(H)ge n^{Omega(log n/loglog n)}$, showing that these Ramsey numbers can grow faster than any polynomial in the number of vertices.
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