On the ramsey numbers N(3,3,…3;2)

AbstractThe main results of this paper are N(3,3,3,3;2) > 50 and f(k+1)≥3 f(k)+f(k−2), where f(k) = N3,3,…;2)ktimes −1 for k ≥ 3

Lower bounds for multi-colored Ramsey numbers from group orbits

In this paper the algorithm developed in [RK] for 2-color Ramsey numbers is generalized to multi-colored Ramsey numbers. All the cyclic graphs yielding the lower bounds R(3,3,4)\u3e=30, R(3,3,5)\u3e=45, and R(3,4,4)\u3e=55 were obtained. The two last bounds are apparently new

Bounds on some van der Waerden numbers

Abstract For positive integers s and k 1 , k 2 , . . . , k s , the van der Waerden number w(k 1 , k 2 , . . . , k s ; s) is the minimum integer n such that for every s-coloring of set {1, 2, . . . , n}, with colors 1, 2, . . . , s, there is a k i -term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k, m; 2) for fixed m. We include a table of values of w(k, 3; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w(k, k, . . . , k; s) that slightly improves previously-known bounds. Upper bounds for w(k, 4; 2) and w(4, 4, . . . , 4; s) are also provided

An algorithmic approach for multi-color Ramsey graphs

The classical Ramsey number R(r1,r2,...,rm) is defined to be the smallest integer n such that no matter how the edges of Kn are colored with the m colors, 1, 2, 3, . . . ,m, there exists some color i such that there is a complete subgraph of size ri, all of whose edges are of color i. The problem of determining Ramsey numbers is known to be very difficult and is usually split into two problems, finding upper and lower bounds. Lower bounds can be obtained by the construction of a, so called, Ramsey graph. There are many different methods to construct Ramsey graphs that establish lower bounds. In this thesis mathematical and computational methods are exploited to construct Ramsey graphs. It was shown that the problem of checking that a graph coloring gives a Ramsey graph is NP-complete. Hence it is almost impossible to find a polynomial time algorithm to construct Ramsey graphs by searching and checking. Consequently, a method such as backtracking with good pruning techniques should be used. Algebraic methods were developed to enable such a backtrack search to be feasible when symmetry is assumed. With the algorithm developed in this thesis, two new lower bounds were established: R(3,3,5)≥45 and R(3,4,4)≥55. Other best known lower bounds were matched, such as R(3,3,4)≥30. The Ramsey graphs giving these lower bounds were analyzed and their full symmetry groups were determined. In particular it was shown that there are unique cyclic graphs up to isomorphism giving R(3,3,4)≥30 and R(3,4,4)≥55, and 13 non-isomorphic cyclic graphs giving R(3,3,5)≥45

Period adding structure in a 2D discontinuous model of economic growth

We study the dynamics of a growth model formulated in the tradition of Kaldor and Pasinetti where the accumulation of the ratio capital/workers is regulated by a two-dimensional discontinuous map with triangular structure. We determine analytically the border collision bifurcation boundaries of periodicity regions related to attracting cycles, showing that in a two-dimensional parameter plane these regions are organized in the period adding structure. We show that the cascade of flip bifurcations in the base one-dimensional map corresponds for the two-dimensional map to a sequence of pitchfork and flip bifurcations for cycles of even and odd periods, respectively

On a Class of Vertex Folkman Numbers

Let a1 , . . . , ar, be positive integers, i=1 ... r, m = ∑(ai − 1) + 1 and p = max{a1 , . . . , ar }. For a graph G the symbol G → (a1 , . . . , ar ) means that in every r-coloring of the vertices of G there exists a monochromatic ai -clique of color i for some i ∈ {1, . . . , r}. In this paper we consider the vertex Folkman numbers F (a1 , . . . , ar ; m − 1) = min |V (G)| : G → (a1 , . . . , ar ) and Km−1 ⊂ G} We prove that F (a1 , . . . , ar ; m − 1) = m + 6, if p = 3 and m ≧ 6 (Theorem 3) and F (a1 , . . . , ar ; m − 1) = m + 7, if p = 4 and m ≧ 6 (Theorem 4)

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a