A Ramsey Treatment of Symmetry

2011

We survey some principal results and open problems related to colorings of geometric and algebraic objects endowed with symmetries, concentrating the exposition on the maximal symmetry numbers of such objects.

Graphs and Combinatorics, 2012

The chromatic number of a subset of the real plane is the smallest number of colors assigned to the elements of that set such that no two points at distance 1 receive the same color. It is known that the chromatic number of the plane is between 4 and 7. In this note, we determine the bounds on the chromatic number for several classes of subsets of the plane such as extensions of the rational plane, sets in convex position, infinite strips, and parallel lines. √ 3 2 forms a subgraph of this graph. Is it true that χ(Q × R, 1) = 4? Is it true that χ(Q × R, 1) < χ(R 2 , 1)? While we cannot answer these questions, we provide some partial results.

2009

We survey some principal results and open problems related to colorings of algebraic and geometric objects endowed with symmetries.

Graphs and Combinatorics, 2014

For given graphs G 1 and G 2 , the Ramsey number R(G 1 , G 2) is the least integer n such that every 2-coloring of the edges of K n contains a subgraph isomorphic to G 1 in the first color or a subgraph isomorphic to G 2 in the second color. Surahmat et al. proved that the Ramsey number R(C 4 , W n) ≤ n+ (n−1)/3. By using asymptotic methods one can obtain the following property: R(C 4 , W n) ≤ n + √ n + o(1). In this paper we show that in fact R(C 4 , W n) ≤ n+ √ n − 2+1 for n ≥ 11. Moreover, by modification of the Erdős-Rényi graph we obtain an exact value R(C 4 , W q 2 +1) = q 2 +q+1 with q ≥ 4 being a prime power. In addition, we provide exact values for Ramsey numbers R(C 4 , W n) for 14 ≤ n ≤ 17.

The Electronic Journal of Combinatorics

We will prove that $R(C_4, C_4, K_4-e)=16$. This fills one of the gaps in the tables presented in a 1996 paper by Arste et al. Moreover by using computer methods we improve lower and upper bounds for some other multicolor Ramsey numbers involving quadrilateral $C_4$. We consider $3$ and $4$-color numbers, our results improve known bounds.

arXiv (Cornell University), 2020

Problems with a Point, 2018

Let G n,m be the grid [n]×[m]. G n,m is c-colorable if there is a function χ : G n,m → [c] such that there are no rectangles with all four corners the same color. We ask for which values of n, m, c is G n,m c-colorable? We determine (1) exactly which grids are 2-colorable, (2) exactly which grids are 3-colorable, (2) exactly which grids are 4-colorable. Our main tools are combinatorics and finite fields. Our problem has two motivations: (1) (ours) A Corollary of the Gallai-Witt theorem states that, for all c, there exists W = W (c) such that any c-coloring of [W ] × [W ] has a monochromatic square. The bounds on W (c) are enormous. Our relaxation of the problem to rectangles yields much smaller bounds. (2) Colorings grids to avoid a rectangle is equivalent to coloring the edges of a bipartite graph to avoid a monochromatic K 2,2,. Hence our work is related to bipartite Ramsey Numbers.

Ars Comb., 2009

For graphs G1, G2, · · · , Gm, the Ramsey number R(G1, G2, · · · , Gm) is defined to be the smallest integer n such that anym-coloring of the edges of the complete graphKn must include a monochromatic Gi in color i, for some i. In this note we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral C4, including R(C4,K9) ≤ 32, 19 ≤ R(C4, C4,K4) ≤ 22, 31 ≤ R(C4, C4, C4,K4) ≤ 50, 52 ≤ R(C4,K4,K4) ≤ 72, 42 ≤ R(C4, C4,K3,K4) ≤ 76, and 87 ≤ R(C4, C4,K4,K4) ≤ 179. Supported by the National Natural Science Foundation of China (Grant Nos. 60533010, 60703047 and 60563008), the Basic Research Fund of Guangxi Academy of Sciences (080414), the Program for New Century Excellent Talents in University (NCET05-0612) and the Chenguang Program of Wuhan (200750731262).

Journal of Combinatorial Theory, Series A, 1992

Discrete Mathematics, 1996

A six-coloring of the Euclidean plane is constructed such that the distance 1 is not realized by any color except one, which does not realize the distance x/2 -1.