For any two-colouring of the segments determined by 3n− 3 points in general position in the plane, either the first colour class contains a triangle, or there is a noncrossing cycle of length n in the second colour class, and this result is tight. We also give a series of more general estimates on off-diagonal geometric graph Ramsey numbers in the same spirit. Finally we investigate the existen...

The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H. The star-critical Ramsey number r∗(G,H) is the smallest integer k such that every 2-coloring of the edges of Kr −K1,r−1−k contains either a red copy of G or a blue copy of H. We will classify the critical graphs, 2-colorings of the complete ...

R. H. Schelp conjectured that if G is a graph with |V (G)| = R(Pn, Pn) such that δ(G) > 3|V (G)| 4 , then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first rep...

An online Ramsey game (G, H) is a game between Builder and Painter, alternating in turns. In each round Builder draws an edge and Painter colors it either red or blue. Builder wins if after some round there is a monochromatic copy of the graph H, otherwise Painter is the winner. The rule for Builder is that after each his move the resulting graph must belong to the class of graphs G. In this ab...

We study an ordered version of hypergraph Ramsey numbers using linearly ordered vertex sets, due to Fox, Pach, Sudakov, and Suk. In the k-uniform ordered path, the edges are the sets of k consecutive vertices in a linear vertex order. Moshkovitz and Shapira described its ordered Ramsey number as the solution to an enumeration problem involving higher-order mulidimensional integer partitions. We...

In this paper, we study an analogue of size-Ramsey numbers for vertex colorings. For a given number of colors r and a graph G the vertex size-Ramsey number of G, denoted by R̂v(G, r), is the least number of edges in a graph H with the property that any r-coloring of the vertices of H yields a monochromatic copy of G. We observe that Ωr(∆n) = R̂v(G, r) = Or(n ) for any G of order n and maximum deg...

An n-vertex graph G is c-Ramsey if it contains neither a complete nor an empty induced subgraph of size greater than c log n. Erdős, Faudree and Sós conjectured that every c-Ramsey graph with n vertices contains Ω(n5/2) induced subgraphs any two of which differ either in the number of vertices or in the number of edges, i.e. the number of distinct pairs (|V (H)|, |E(H)|), as H ranges over all i...