Sharp thresholds for the phase transition between primitive recursive and Ackermannian Ramsey numbers

We compute the sharp thresholds on g at which g-large and g-regressive Ramsey numbers cease to be primitive recursive and become Ackermannian.We also identify the threshold below which g-regressive colorings have usual Ramsey numbers, that is, admit homogeneous, rather than just min-homogeneous sets.     

A classification of rapidly growing Ramsey functions

Let be a number-theoretic function. A finite set of natural numbers is called -large if . Let be the Paris Harrington statement where we replace the largeness condition by a corresponding -largeness condition. We classify those functions for which the statement is independent of first order (Peano) arithmetic . If is a fixed iteration of the binary length function, then is independent. On the other hand is provable in . More precisely let where denotes the -times iterated binary length of and denotes the inverse function of the -th member of the Hardy hierarchy. Then is independent of (for ) iff .

     

Upper bounds on probability thresholds for asymmetric Ramsey properties

Given two graphs G and H, we investigate for which functions the random graph (the binomial random graph on n vertices with edge probability p) satisfies with probability that every red‐blue‐coloring of its edges contains a red copy of G or a blue copy of H. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes the exact order of magnitude of the threshold for the case in which G and H are complete graphs of arbitrary size. In our proof we present an alternative to the so‐called deletion method, which was introduced by Rödl and Ruciński in their study of symmetric Ramsey properties of random graphs (i.e. the case G = H), and has been used in many proofs of similar results since then.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 1–28, 2014     

Constrained Ramsey numbers for rainbow matching

The Ramsey number R_k(G) of a graph G is the minimum number N, such that any edge coloring of K_N with k colors contains a monochromatic copy of G. The constrained Ramsey number f(G, T) of the graphs G and T is the minimum number N, such that any edge coloring of K_N with any number of colors contains a monochromatic copy of G or a rainbow copy of T. We show that these two quantities are closely related when T is a matching. Namely, for almost all graphs G, f(G, tK₂) = R_{t ? 1}(G) for t≥2. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:91‐95, 2011     

Upper bounds for Ramsey numbers

The Ramsey number R(G₁,G₂,…,G_k) is the least integer p so that for any k-edge coloring of the complete graph K_p, there is a monochromatic copy of G_i of color i. In this paper, we derive upper bounds of R(G₁,G₂,…,G_k) for certain graphs G_i. In particular, these bounds show that R(9,9)6588 and R(10,10)23556 improving the previous best bounds of 6625 and 23854.     

Multipartite Ramsey numbers for odd cycles

In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n≥5 be an arbitrary positive odd integer; then, in any two‐coloring of the edges of the complete 5‐partite graph K((n?1)/2, (n?1)/2, (n?1)/2, (n?1)/2, 1) there is a monochromatic C_n, a cycle of length n. This roughly says that the Ramsey number for C_n (i.e. 2n?1 ) will not change (somewhat surprisingly) if four large “holes” are allowed. Note that this would be best possible as the statement is not true if we delete from K_2n?1 the edges within a set of size (n+ 1)/2. We prove an approximate version of the above conjecture. © 2009 Wiley Periodicals, Inc. J Graph Theory 61:12‐21, 2009     

The Ramsey number for hypergraph cycles I

Let C_n denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v₁,…,v_n and edges v₁v₂v₃, v₃v₄v₅, v₅v₆v₇,…,v_n-1v_nv₁. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C_n, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible.     

Tripartite Ramsey numbers for paths

In this article, we study the tripartite Ramsey numbers of paths. We show that in any two‐coloring of the edges of the complete tripartite graph K(n, n, n) there is a monochromatic path of length (1 ? o(1))2n. Since R(P_2n+1,P_2n+1)=3n, this means that the length of the longest monochromatic path is about the same when two‐colorings of K_3n and K(n, n, n) are considered. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 164–174, 2007     

Degree Ramsey numbers for even cycles

Let $H\stackrel{s}{?}G$ denote that any $s$-coloring of $E\left(H\right)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_{\Delta }(G,s)$, is $min\{\Delta \left(H\right):H\stackrel{s}{?}G\}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_{\Delta }(C_{2k},s)\ge 2s$, and Kang and Perarnau showed that $R_{\Delta }(C_4,s)=\Theta \left(s^2\right)$. Our main result is that $R_{\Delta }(C_6,s)=\Theta \left(s^{3∕2}\right)$ and $R_{\Delta }(C_{10},s)=\Theta \left(s^{5∕4}\right)$. Additionally, we substantially improve the lower bound for $R_{\Delta }(C_{2k},s)$ for general $k$.     

Gallai–Ramsey numbers for cycles

For given graphs G and H and an integer k, the Gallai–Ramsey number is defined to be the minimum integer n such that, in any k coloring of the edges of K_n, there exists a subgraph isomorphic to either a rainbow coloring of G or a monochromatic coloring of H. In this work, we consider Gallai–Ramsey numbers for the case when G=K₃ and H is a cycle of a fixed length.     

Size Ramsey numbers for some regular graphs

P. Erdös, R.J. Faudree, C.C. Rousseau and R.H. Schelp [P. Erdös, R.J. Faudree, C.C. Rousseau, R.H. Schelp, The size Ramsey number, Period. Math. Hungar. 9 (1978) 145-161] studied the asymptotic behaviour of for certain graphs G,H. In this paper there will be given a lower bound for the diagonal size Ramsey number of K_n,n,n. The result is a generalization of a theorem for K_n,n given by P. Erdös and C.C. Rousseau [P. Erdös, C.C. Rousseau, The size Ramsey numbers of a complete bipartite graph, Discrete Math. 113 (1993) 259-262].Moreover, an open question for bounds for size Ramsey number of each n-regular graph of order n+t for t>n−1 is posed.     

A note on Ramsey numbers for books

The book with n pages B_n is the graph consisting of n triangles sharing an edge. The book Ramsey number r(B_m,B_n) is the smallest integer r such that either B_m ? G or B_n ? G for every graph G of order r. We prove that there exists a positive constant c such that r(B_m,B_n) = 2n + 3 for all n ≥ cm. Our proof is based mainly on counting; we also use a result of Andrásfai, Erd?s, and Sós stating that triangle‐free graphs of order n and minimum degree greater than 2n/5 are bipartite. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Star-critical Ramsey numbers for large generalized fans and books

In this paper, we show that for any fixed integers $m\ge 2$ and $t\ge 2$, the star-critical Ramsey number $r_1(K_1+n{K}_t,K_{m+1})=(m?1)tn+t$ for all sufficiently large $n$. Furthermore, for any fixed integers $p\ge 2$ and $m\ge 2$, $r_1(K_p+n{K}_1,K_{m+1})=(m?1+o\left(1\right))n$ as $n\to \infty$.     

Some geometric structures and bounds for Ramsey numbers

We investigate several bounds for both K_2,m−K_1,n Ramsey numbers and K_2,m−K_1,n bipartite Ramsey numbers, extending some previous results. Constructions based on certain geometric structures (designs, projective planes, unitals) yield classes of near-optimal bounds or even exact values. Moreover, relationships between these numbers are also discussed.     

The tail is cut for Ramsey numbers of cubes

The Ramsey number R(G) of a graph G is the least integer p such that for all bicolorings of the edges of the complete graph K_p, one of the monochromatic subgraphs contains a copy of G. We show that for any positive constant c and bipartite graph G=(U,V;E) of order n where the maximum degree of vertices in U is at most , . Moreover, we show that the Ramsey number of the cube Q_n of dimension n satisfies . In both cases, the small terms are removed from the powers in the upper bounds of a earlier result of the author.     

The Ramsey numbers for disjoint unions of trees

For given graphs G and H, the Ramsey numberR(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper, we investigate the Ramsey number R(∪G,H), where G is a tree and H is a wheel W_m or a complete graph K_m. We show that if n?3, then R(kS_n,W₄)=(k+1)n for k?2, even n and R(kS_n,W₄)=(k+1)n-1 for k?1 and odd n. We also show that .     

New upper bound formulas with parameters for Ramsey numbers

In this paper, we obtain some new results R(5,12)?848, R(5,14)?1461, etc., and we obtain new upper bound formulas for Ramsey numbers with parameters.     

CO‐irredundant Ramsey numbers for graphs

The study of the CO‐irredundant Ramsey numbers t(n₁, ···, n_k) is initiated. It is shown that several values and bounds for these numbers may be obtained from the well‐studied generalized graph Ramsey numbers and the values of t(4, 5), t(4, 6) and t(3, 3, m) are calculated. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 258–268, 2000     

The Ramsey numbers for disjoint unions of graphs

For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper we investigate the Ramsey number of a disjoint union of graphs . For any natural integer k, we contain a general upper bound, R(kG,H)?R(G,H)+(k-1)|V(G)|. We also show that if m=2n-4, 2n-8 or 2n-6, then R(kS_n,W_m)=R(S_n,W_m)+(k-1)n. Furthermore, if |G_i|>(|G_i|-|G_i+1|)(χ(H)-1) and R(G_i,H)=(χ(H)-1)(|G_i|-1)+1, for each i, then .