Induced Ramsey Numbers Research Articles

Induced Ramsey Number for a Star Versus a Fixed Graph

We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let $\mathrm{IR}(H,G)$ be the smallest number of vertices in a graph $F$ such that $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$.
 In this note we consider the case when $G$ is a star on $n$ edges, for large $n$ and $H$ is a fixed graph. We prove that $$ (\chi(H)-1) n \leq \mathrm{IR}(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$, sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.

On induced Ramsey numbers for multiple copies of graphs

AbstractWe say that a graph strongly arrows a pair of graphs and write if any coloring of its edges with red and blue leads to either a red or a blue appearing as induced subgraphs of . The induced Ramsey number, , is defined as . We consider the connection between the induced Ramsey number for a pair of two connected graphs and the induced Ramsey number for multiple copies of these graphs , where denotes the pairwise vertex‐disjoint union of copies of . It is easy to see that if , then . This implies that For several specific graphs, such as a path on three vertices vs a complete multipartite graph, a matching vs a complete graph, or a matching vs another matching, it is known that the above inequality is tight. On the other hand, the inequality is also strict for some graphs. However, even in the simplest case when is an edge and , it is not known for what and for what the above inequality is tight. We show that it is tight if is connected and is at least as large as the order of . In addition, we make further progress in determining induced Ramsey numbers for multiple copies of graphs, such as paths and triangles.

On lower bound for induced Ramsey numbers

On lower bound for induced Ramsey numbers

On induced Ramsey numbers fork-uniform hypergraphs

Let denote the complete k-uniform k-partite hypergraph with classes of size t and the complete k-uniform hypergraph of order s. One can show that the Ramsey number for and satisfies when t = so(1) as s →∞. The main part of this paper gives an analogous result for induced Ramsey numbers: Let be an arbitrary k-partite k-uniform hypergraph with classes of size t and an arbitrary k-graph of order s. We use the probabilistic method to show that the induced Ramsey number (i.e. the smallest n for which there exists a hypergraph such that any red/blue coloring of yields either an induced red copy of or an induced blue copy of ) satisfies . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 5–20, 2016

On two problems in graph Ramsey theory

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph K N contains a monochromatic copy of H. A famous result of Chváatal, Rödl, Szemerédi and Trotter states that there exists a constant c(Δ) such that r(H) ≤ c(Δ)n for every graph H with n vertices and maximum degree Δ. The important open question is to determine the constant c(Δ). The best results, both due to Graham, Rödl and Ruciński, state that there are positive constants c and c′ such that $$2^{c'\Delta } \leqslant c(\Delta ) \leqslant ^{c\Delta \log ^2 \Delta }$$ . We improve this upper bound, showing that there is a constant c for which c(Δ) ≤ 2 cΔlogΔ . The induced Ramsey number r ind (H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erdős conjectured the existence of a constant c such that, for any graph H on n vertices, r ind (H) ≤ 2 cnlogn . We move a step closer to proving this conjecture, showing that r ind (H) ≤ 2 cnlogn . This improves upon an earlier result of Kohayakawa, Prömel and Rödl by a factor of logn in the exponent.

Induced Ramsey-type theorems

We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rödl, Erdős–Hajnal, Prömel–Rödl, Nikiforov, Chung–Graham, and Łuczak–Rödl. The proofs are based on a simple lemma (generalizing one by Graham, Rödl, and Ruciński) that can be used as a replacement for Szemerédi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.

Induced Ramsey-type theorems

Abstract We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Łuczak-Rodl, Promel-Rodl, Erdős-Hajnal, and Nikiforov. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.

On induced Ramsey numbers

The induced Ramsey number IR( G, H) is defined as the smallest integer n, for which there exists a graph F on n vertices such that any 2-colouring of its edges with red and blue leads to either a red copy of G induced in F, or an induced blue H. In this note, we study the value of the induced Ramsey numbers, as well as their planar and weak versions, for some special classes of graphs. In particular, we show that, for the induced planar Ramsey numbers, the fact whether we prohibit monochromatic copies induced in the graph, or induced just in its own colour, may significantly affect the value of the Ramsey number.

A note on a triangle-free — complete graph induced Ramsey number

The induced Ramsey number, IR( G, H), is equal to p if there exists a graph F on p vertices such that any 2-colouring of its edges with red and blue leads to either an induced copy G in the subgraph of F spanned by the red edges or an induced blue H, and, furthermore, no graphs on p−1 vertices have the above property. There will be shown that the lower bound of the induced Ramsey number for a triangle-free graph on t vertices and a complete graph K n is roughly n 2 t/4. In one case, when the triangle-free graph is a star, a simple proof of the exact value (about n 2 t/2) will be given.

Induced Ramsey Numbers

We investigate the induced Ramsey number of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with , we have where is the chromatic number of H and C is some universal constant. Furthermore, we also investigate imposing some conditions on G. For instance, we prove a bound that is polynomial in both k and t in the case in which G is a tree. Our methods of proof employ certain random graphs based on projective planes.

On Induced Ramsey Numbers for Graphs with Bounded Maximum Degree

For graphs G and H we write G → ind H if every 2-edge colouring of G yields an induced monochromatic copy of H . The induced Ramsey number for H is defined as r ind ( H )=min{| V ( G )|: G → ind H }. We show that for every d ⩾1 there exists an absolute constant c d such that r ind ( H n, d )⩽ n c d for every graph H n, d with n vertices and the maximum degree at most d . This confirms a conjecture suggested by W. T. Trotter.