Learn More
Proof. Let G satisfy the hypothesis of Theorem 1. Clearly, G contains a circuit ; let C be the longest one . If G has no Hamiltonian circuit, there is a vertex x with x ~ C . Since G is s-connected, there are s paths starting at x and terminating in C which are pairwise disjoint apart from x and share with C just their terminal vertices x l, X2, . . ., x s(More)
In this paper we will consider Ramsey-type problems for finite graphs, r-partitions and hypergraphs. All these problems ask for the existence of large homogeneous (monochromatic) configurations of a certain kind under the condition that the size of the underlying set is large. As it is quite common in Ramsey theory, most of our results are not sharp and(More)
Let H be a fixed graph of chromatic number r. It is shown that the number of graphs on n vertices and not containing H as a subgraph is 2 ( z)( '-•'i * O( ')). Let h,(n) denote the maximum number of edges in an r-uniform hypergraph on n vertices and in which the union of any three edges has size greater than 3r 3. It is shown that h,(n) = o(n 2 ) although(More)
A family of sets {Ak i is said to have property B if there is a set S which meets every A k and contains none of them . m(n) is the smallest integer so that there is a family {A k }, 1 < k < m(n), I A k = n which does not have property B . It is known that [2, 3, 6] 2n (1 + nl < m(n) < cn2 2 71 . in(2) = 3 ; m(3) = 7; m(4) is not known . Now we define a(More)
Assume that K is a finite complete graph whose edges are colored with r colors (r ~ 2 ). How many monochromatic paths (or cycles) are needed to cover (or partition) the vertex set of K? Throughout the paper single vertices and edges are considered to be cycles. It is not obvious that these numbers depend only on r. The following conjecture is from [ 12]. If(More)
It is easy to see that (2) implies that if g(z) is not a polynomial and the order of f(z) is positive then the order of f(g(z)) must be infinite and Bakers result shows that at least if the orders of f(z) and g(z) are less than 1 Pólyas result can not be strengthened, since u(r) can tend to infinity as slowly as we please . In the present note we are going(More)
We shall consider graphs (hypergraphs) without loops and multiple edges . Let Ybe a family of so called prohibited graphs and ex (n, Y) denote the maximum number of edges (hyperedges) a graph (hypergraph) on n vertices can have without containing subgraphs from Y A graph (hypergraph) will be called supersaturated if it has more edges than ex (n, Y). If G(More)