Search
Filters
total: 19
Search results for: RAMSEY NUMBERS
-
Polyhedral Ramsey Numbers
PublicationGiven two polygons or polyhedrons P1 and P2, we can transform these figures to graphs G1 and G2, respectively. The polyhedral Ramsey number Rp(G1,G2) is the smallest integer n such that every graph, which represents polyhedron on n vertices either contains a copy of G1 or its complement contains a copy of G2. Using a computer search together with some theoretical results we have established some polyhedral Ramsey numbers, for example...
-
Shannon Capacity and Ramsey Numbers
PublicationRamsey-type theorems are strongly related to some results from information theory. In this paper we present these relations.
-
On some Zarankiewicz numbers and bipartite Ramsey Numbers for Quadrilateral
PublicationThe Zarankiewicz number z ( m, n ; s, t ) is the maximum number of edges in a subgraph of K m,n that does not contain K s,t as a subgraph. The bipartite Ramsey number b ( n 1 , · · · , n k ) is the least positive integer b such that any coloring of the edges of K b,b with k colors will result in a monochromatic copy of K n i ,n i in the i -th color, for some i , 1 ≤ i ≤ k . If n i = m for all i , then we denote this number by b k ( m )....
-
A NOTE ON ON-LINE RAMSEY NUMBERS FOR QUADRILATERALS
PublicationWe consider on-line Ramsey numbers defined by a game played between two players, Builder and Painter. In each round Builder draws an the edge and Painter colors it either red or blue, as it appears. Builder’s goal is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number \widetilde{r}(H) of...
-
On-line Ramsey Numbers of Paths and Cycles
PublicationConsider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy of $H$ for some fixed graphs $G$ and $H$. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the \emph{on-line Ramsey number} $\tilde{r}(G,H)$. In...
-
Ramsey numbers for triangles versus almost-complete graphs.
PublicationPokazano, że w każdym krawędziowym pokolorowaniu dwoma kolorami grafu pełnego o 38 wierzchołkach występuje trójkąt w pierwszym kolorze lub podgraf izomorficzny z K_10 - e w drugim kolorze. Stąd otrzymujemy górne oszacowanie R(K_3, K_10 - e) <= 38. Przedstawiamy także pokolorowanie krawędziowe grafu K_36, którego istnienie dowodzi, że R(K_3, K_10 - e) >= 37.
-
On some open questions for Ramsey and Folkman numbers
PublicationWe discuss some of our favorite open questions about Ramsey numbers and a related problem on edge Folkman numbers. For the classical two-color Ramsey numbers, we first focus on constructive bounds for the difference between consecutive Ramsey numbers. We present the history of progress on the Ramsey number R(5,5) and discuss the conjecture that it is equal to 43.
-
On some ramsey and turan-type numbers for paths and cycles
PublicationUdowodniono, że R(P_3,C_k,C_k)= R(C_k,C_k)= 2k - 1, dla nieparzystych k. Udowodniono, że R(P_4,P_4,C_k) = k + 2 oraz R(P_3,P_5,C_k) = k + 1 dla k > 2.
-
A survey on known values and bounds on the Shannon capacity
PublicationIn this survey we present exact values and bounds on the Shannon capacity for different classes of graphs, for example for regular graphs and Kneser graphs. Additionally, we show a relation between Ramsey numbers and Shannon capacity.
-
Dataset of non-isomorphic graphs of the coloring types (K4,K4;n), 1<n<R(4,4)
Open Research DataFor K4 graph, a coloring type (K4,K4;n) is such an edge coloring of the full Kn graph, which does not have the K4 subgraph in the first color (representing by no edges in the graph) or the K4 subgraph in the second color (representing by edges in the graph).The Ramsey number R(4,4) is the smallest natural number n such that for any edge coloring of...
-
Liczby Ramseya on-line dla różnych klas grafów
PublicationRozpatrujemy grę rozgrywaną na nieskończonej liczbie wierzchołków, w której każda runda polega na wskazaniu krawędzi przez jednego gracza - Budowniczego oraz pokolorowaniu jej przez drugiego gracza - Malarkę na jeden z dwóch kolorów, czerwony lub niebieski. Celem Budowniczego jest zmuszenie Malarki do stworzenia monochromatycznej kopii wcześniej ustalonego grafu H w jak najmniejszej możliwej liczbie ruchów. Zakładamy, że gracze...
-
Dataset of non-isomorphic graphs being coloring types (K5-e,Km-e;n), 2<m<5, 1<n<R(K5-e,Km-e)
Open Research DataFor K5-e and Km-e graphs, the type coloring (K5-e,Km-e;n) is such an edge coloring of the full Kn graph, which does not have the K5-e subgraph in the first color (no edge in the graph) or the Km-e subgraph in the second color (exists edge in the graph). Km-e means the full Km graph with one edge removed.The Ramsey number R(K5-e,Km-e) is the smallest...
-
Dataset of non-isomorphic graphs being coloring types (K6-e,Km-e;n), 2<m<5, 1<n<R(K6-e,Km-e)
Open Research DataFor K6-e and Km-e graphs, the type coloring (K6-e,Km-e;n) is such an edge coloring of the full Kn graph, which does not have the K6-e subgraph in the first color (no edge in the graph) or the Km-e subgraph in the second color (exists edge in the graph). Km-e means the full Km graph with one edge removed. The Ramsey number R(K6-e,Km-e) is the smallest...
-
Dataset of non-isomorphic graphs of the coloring types (K3,Km;n), 2<m<7, 1<n<R(3,m)
Open Research DataFor K3 and Km graphs, a coloring type (K3,Km;n) is such an edge coloring of the full Kn graph, which does not have the K3 subgraph in the first color (representing by no edges in the graph) or the Km subgraph in the second color (representing by edges in the graph).The Ramsey number R(3,m) is the smallest natural number n such that for any edge coloring...
-
Dataset of non-isomorphic graphs of the coloring types (Km,K3-e;n), 4<m<8, 1<n<R(Km,K3-e)
Open Research DataFor Km and K3-e graphs, a coloring type (Km,K3-e;n) is such an edge coloring of the full Kn graph, which does not have the Km subgraph in the first color (representing by no edges in the graph) or the K3-e subgraph in the second color (representing by edges in the graph). K3-e means the full Km graph with one edge removed.The Ramsey number R(Km,K3-e)...
-
Dataset of non-isomorphic graphs of the coloring types (K3,Km-e;n), 2<m<7, 1<n<R(K3,Km-e).
Open Research DataFor K3 and Km-e graphs, a coloring type (K3,Km-e;n) is such an edge coloring of the full Kn graph, which does not have the K3 subgraph in the first color (representing by no edges in the graph) or the Km-e subgraph in the second color (representing by edges in the graph). Km-e means the full Km graph with one edge removed.The Ramsey number R(K3,Km-e)...
-
Dataset of non-isomorphic graphs of the coloring types (K4,Km-e;n), 2<m<5, 1<n<R(K4,Km-e)
Open Research DataFor K4 and Km-e graphs, a coloring type (K4,Km-e;n) is such an edge coloring of the full Kn graph, which does not have the K4 subgraph in the first color (representing by no edges in the graph) or the Km-e subgraph in the second color (representing by edges in the graph). Km-e means the full Km graph with one edge removed.The Ramsey number R(K4,Km-e)...
-
Dataset of non-isomorphic graphs being coloring types (K4-e,Km-e;n), 2<m<7, 1<n<R(K4-e,Km-e)
Open Research DataFor K4-e and Km-e graphs, the type coloring (K4-e,Km-e;n) is such an edge coloring of the full Kn graph, which does not have the K4-e subgraph in the first color (no edge in the graph) or the Km-e subgraph in the second color (exists edge in the graph). Km-e means the full Km graph with one edge removed.The Ramsey number R(K4-e,Km-e) is the smallest...
-
Dataset of non-isomorphic graphs being coloring types (K3-e,Km-e;n), 2<m<8, 1<n<R(K3-e,Km-e)
Open Research DataFor K3-e and Km-e graphs, the type coloring (K3-e,Km-e;n) is such an edge coloring of the full Kn graph, which does not have the K3-e subgraph in the first color (no edge in the graph) or the Km-e subgraph in the second color (exists edge in the graph). Km-e means the full Km graph with one edge removed.The Ramsey number R(K3-e,Km-e) is the smallest...