Stefan A. Burr

Generalizations of a Ramsey‐theoretic result of chvátal

Chvatal has shown that ifT is a tree on n points then r(Kk, T) _ (k- 1 ) (n- 1 )+ 1,where r is the (generalized) Ramsey number . It is shown that the same result holds whenT is replaced by many other graphs . Such a T is called kgood. The results proved all support the conjecture that any large graph that is sufficiently sparse, in the appropriate sense, is k-good .

On the Ramsey multiplicities of graphs—problems and recent results

Ramseys theorem guarantees that if G is a graph, then any 2-coloring of the edges of a large enough complete graph yields a monochromatic copy of G. Interesting problems arise when one asks how many such G must occur. A survey of this and related problems is given, along with a number of new results.

Integer Prim-Read Solutions to a Class of Target Defense Problems

We study the choice of a deployment and firing doctrine for defending separated point targets of potentially different values against an attack by an unknown number of sequentially arriving missiles. We minimize the total number of defenders, subject to an upper bound on the maximum expected value damage per attacking weapon. We show that the Greedy Algorithm produces an optimal integral solution to this problem.

A SURVEY OF NONCOMPLETE RAMSEY THEORY FOR GRAPHS

In recent years, the study of generalized Ramsey theory for graphs has grown to become a sizable and sprawling field. In the case of generalized Ramsey numbers and related problems, surveys have been made [4, 47, 481. It seems timely, then, to survey another part of the field. In this paper we shall study what may be called “noncomdete Ramsey theory” in that the graphs to be Golored are not, in general, complete. To clarify this, we must make some definitions. Generally, notation not defined here follows that of Harary [46], except that our graphs will have no isolates. In particular, we distinguish between subgraphs and induced subgraphs.

Some Complete Bipartite Graph - Tree Ramsey Numbers

We investigate r(Ka,a,, T) for a = 2 and a = 3, where T is an arbitrary tree of order n. For a = 2, this Ramsey number is completely determined by r(K2,2, K1,m) where m = Δ(T). For a = 3, we do not find such an “exact” result, but we do show that r(K3,3, T)

On the use of senders in generalized ramsey theory for graphs

Abstract If F , G and H are graphs, write F → ( G , H ) to mean that however the edges of F are colored red and blue, either the red (partial) subgraph contains a copy of G or the blue subgraph contains a copy of H . Many interesting questions exist concerning this relation, particularly involving the case in which F is minimal for this property. A useful tool for constructing graphs relevant to such questions, at least when G and H are 3-connected, is developed here, namely graphs called senders . These senders are used to prove a number of theorems about the class of minimal F , as well as various related results. For example, let each of G and H be 3-connected, or a triangle. Then there exists an α > 0 such that if n is sufficiently large, there are at least e α n log n nonisomorphic F such that F → ( G , H ) in a minimal way.

Ramsey numbers for the pair sparse graph-path or cycle

Introduction. Let G and H be simple graphs. The Ramsey number r(G, H) is the smallest integer n such that for each graph F on n vertices, either G is a subgraph of F or H is a subgraph of F, the complement of F. Calculation of r(G, H) for particular pairs of graphs G and H has received considerable attention, and a survey of such results can be found in [2]. Chvatal [5] proved that if Tn is a tree on n vertices and Km is a complete graph on m vertices, then r(T7,, Kin) = (n 1)(m 1) + 1. In [4] it was shown that if Tn is replaced by a sparse connected graph Gn on n vertices the Ramsey number remains the same (i.e. r(G,, Kin) = (n 1)(m 1) + 1). For m = 3 Chvatals theorem implies r(T1, K3) = 2n -1. In this paper we will show that if Tn is replaced by any sparse connected graph G on n vertices and K3 is replaced by an odd cycle Ck, then for appropriate n the Ramsey number is unchanged. In particular we will prove the following.

Diagonal Ramsey numbers for small graphs

The (generalized) Ramsey number r(G) is determined for all 113 graphs with no more than six lines and no isolated points. While few proofs are given, information is given which should be sufficient to reconstruct them in most cases.

The Unreasonable effectiveness of number theory

The unreasonable effectiveness of number theory in physics, communication, and music by M. R. Schroeder The reasonable and unreasonable effectiveness of number theory in statistical mechanics by G. E. Andrews Number theory and dynamical systems by J. C. Lagarias The mathematics of random number generators by G. Marsaglia Cyclotomy and cyclic codes by V. Pless Number theory in computer graphics by M. D. McIlroy.

On Ramsey numbers for large disjoint unions of graphs

Abstract Let G be a fixed finite set of connected graphs. Results are given which, in principle, permit the Ramsey number r ( G , H ) to be evaluated exactly when G and H are sufficiently large disjoint unions of graphs taken from G . Such evaluations are often possible in practice, as shown by several examples. For instance, when m and n are large, and m ⩽ n , r(mK k , nK l )=(k − 1)m+ln+r(K k−1 , K l−1 )−2.