On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.Comment: 16 page

Solving Graph Coloring Problems with Abstraction and Symmetry

This paper introduces a general methodology, based on abstraction and symmetry, that applies to solve hard graph edge-coloring problems and demonstrates its use to provide further evidence that the Ramsey number $R(4,3,3)=30$. The number $R(4,3,3)$ is often presented as the unknown Ramsey number with the best chances of being found "soon". Yet, its precise value has remained unknown for more than 50 years. We illustrate our approach by showing that: (1) there are precisely 78{,}892 $(3,3,3;13)$ Ramsey colorings; and (2) if there exists a $(4,3,3;30)$ Ramsey coloring then it is (13,8,8) regular. Specifically each node has 13 edges in the first color, 8 in the second, and 8 in the third. We conjecture that these two results will help provide a proof that no $(4,3,3;30)$ Ramsey coloring exists implying that $R(4,3,3)=30$

Constrained Ramsey Numbers

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision

New Lower Bounds for van der Waerden Numbers Using Distributed Computing

This paper provides new lower bounds for van der Waerden numbers. The number $W(k,r)$ is defined to be the smallest integer $n$ for which any $r$-coloring of the integers $0 \ldots, n-1$ admits monochromatic arithmetic progression of length $k$; its existence is implied by van der Waerden's Theorem. We exhibit $r$-colorings of $0\ldots n-1$ that do not contain monochromatic arithmetic progressions of length $k$ to prove that $W(k, r)>n$. These colorings are constructed using existing techniques. Rabung's method, given a prime $p$ and a primitive root $\rho$, applies a color given by the discrete logarithm base $\rho$ mod $r$ and concatenates $k-1$ copies. We also used Herwig et al's Cyclic Zipper Method, which doubles or quadruples the length of a coloring, with the faster check of Rabung and Lotts. We were able to check larger primes than previous results, employing around 2 teraflops of computing power for 12 months through distributed computing by over 500 volunteers. This allowed us to check all primes through 950 million, compared to 10 million by Rabung and Lotts. Our lower bounds appear to grow roughly exponentially in $k$. Given that these constructions produce tight lower bounds for known van der Waerden numbers, this data suggests that exact van der Waerden Numbers grow exponentially in $k$ with ratio $r$ asymptotically, which is a new conjecture, according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader comment

Vertex Ramsey problems in the hypercube

If we 2-color the vertices of a large hypercube what monochromatic substructures are we guaranteed to find? Call a set S of vertices from Q_d, the d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n sufficiently large, contains a monochromatic copy of S. Ramsey's theorem tells us that for any r \geq 1 every 2-coloring of a sufficiently large r-uniform hypergraph will contain a large monochromatic clique (a complete subhypergraph): hence any set of vertices from Q_d that all have the same weight is Ramsey. A natural question to ask is: which sets S corresponding to unions of cliques of different weights from Q_d are Ramsey? The answer to this question depends on the number of cliques involved. In particular we determine which unions of 2 or 3 cliques are Ramsey and then show, using a probabilistic argument, that any non-trivial union of 39 or more cliques of different weights cannot be Ramsey. A key tool is a lemma which reduces questions concerning monochromatic configurations in the hypercube to questions about monochromatic translates of sets of integers.Comment: 26 pages, 3 figure

Local colourings and monochromatic partitions in complete bipartite graphs

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.Comment: 18 page