We study a combinatorial game called Bichromatic Triangle Game, defined as follows. Two players R and B construct a triangulation on a given planar point set V . Starting from no edges, they take turns drawing one straight edge that connects two points in V and does not cross any of the previously drawn edges. PlayerR uses color red and player B uses color blue. The first player who completes o...

Seismic interpreters are often faced with the need to visualise and interpret a number of seismic attributes simultaneously. Side-by-side visualisation of the individual attributes and use of monochromatic colour bars are still widely used techniques, which nevertheless present a number of limitations. In this paper we discuss how colour can be used effectively in order to support simultaneous ...

We study a combinatorial game called Bichromatic Triangle Game, defined as follows. Two players R and B construct a triangulation on a given planar point set V . Starting from no edges, players R and B take turns drawing one edge that connects two points in V . Player R uses color red and player B uses color blue. The first player who completes one empty monochromatic triangle is the winner. We...

A graph G is called (K3,K3)-co-critical if the edges of G can be coloured with two colours without getting a monochromatic triangle, but adding any new edge to the graph, this kind of ’good’ colouring is impossible. In this short note we construct (K3,K3)-co-critical graphs of maximal degree O(n3/4).

We give a very short proof of the following result of Graham from 1980: For any finite coloring of R, d ≥ 2, and for any α > 0, there is a monochromatic (d + 1)tuple that spans a simplex of volume α. Our proof also yields new estimates on the number A = A(r) defined as the minimum positive value A such that, in any rcoloring of the grid points Z of the plane, there is a monochromatic triangle o...

Recently Papadimitriou has proposed a randomised ‘bit flipping’ method for solving the 2-satisfiability problem; and the author has proposed a randomised recolouring method which, given a 3-colourable graph, finds a 2-colouring of the vertices so that no triangle is monochromatic. Both methods involve finding a ‘bad’ configuration (unsatisfied clause, monochromatic triangle) and randomly changi...

Let F (n, k) denote the maximum number of two edge colorings of a graph on n vertices that admit no monochromatic Kk, (a complete graph on k vertices). The following results are proved: F (n, 3) = 2bn /4c for all n ≥ 6. F (n, k) = 2 k−2 2k−2+o(1))n 2 . In particular, the first result solves a conjecture of Erdös and Rothschild.

In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set {1, . . . , n}, the number Dn is defined as the least number of points the space must contain in order to be sure that there will be an equilat...

The paper gives an account of previous and recent attempts to determine the order of a smallest graph not containing K5 and such that every 2-coloring of its edges results in a monochromatic triangle. A new 14-vertex K4-free graph with the same Ramsey property in the vertex coloring case is found. This yields a new construction of one of the only two known 15-vertex (3,3)-Ramsey graphs not cont...

A digraph D is said to be an m-coloured digraph, if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) of vertices of D is said to be a kernel by monochromatic paths of the m-coloured digraph D, if it satisfies the two following properties: (1) N is independent by monochromatic paths; i.e., f...