The cyclomatic number of connected graphs without solvable orbits

A graph is without solvable orbits if its group of automorphisms acts on each of its orbits through a non-solvable quotient. We prove that there is a connected graph without solvable orbits of cyclomatic number c if and only if c is equal to 6, 8, 10, 11, 15, 16, 19, 20, 21, 22, or is at least 24, and briefly discuss the geometric consequences

On geometric graph Ramsey numbers

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 secondcolour 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 existence of large noncrossing monochromatic matchings in multicoloured geometric graphs

Restricted set addition: The exceptional case of the Erdos-Heilbronn conjecture

Let A,B be different nonempty subsets of the group of integers modulo a prime p. If p is not smaller than |A|+|B|-2, then at least this many residue classes can be represented as a+b, where a and b are different elements of A and B, respectively. This result complements the solution of a problem of Erdos and Heilbronn obtained by Alon, Nathanson, and Ruzsa

Restricted set addition: The exceptional case of the Erdos-Heilbronn conjecture

Let A,B be different nonempty subsets of the group of integers modulo a prime p. If p is not smaller than |A|+|B|-2, then at least this many residue classes can be represented as a+b, where a and b are different elements of A and B, respectively. This result complements the solution of a problem of Erdos and Heilbronn obtained by Alon, Nathanson, and Ruzsa

Balanced subset sums of dense sets of integers

Given n different positive integers not greater than 2n-2, we prove that more than n^2/12 consecutive integers can be represented as the sum of half of the given numbers. This confirms a conjecture of Lev

On geometric graph Ramsey numbers

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 secondcolour 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 existence of large noncrossing monochromatic matchings in multicoloured geometric graphs

Empty convex polygons in almost convex sets

A finite set of points, in general position in the plane, is almost convex if every triple determines a triangle with at most one point in its interior. For every k>2, we determine the maximum size of an almost convex set that does not contain the vertex set of an empty convex k-gon

An algorithm for finding many disjoint monochromatic edges in a complete 2-colored geometric graph

We present an O(n log log n+2 )-time algorithm for finding n disjoint monochromatic edges in a complete geometric graph of 3n - 1 vertices, where the edges are colored by two colors

Ramsey-type results for geometric graphs

We show that for any 2-coloring of the ... segments determined by n points in the plane, one of the color classes contains non-crossing cycles of lengths 3, 4, ..., ⌊√(n/2)⌋. This result is tight up to a multiplicative constant. Under the same assumptions, we also prove that there is a non-crossing path of length &Omega(n&sup2/³), all of whose edges are of the same color. In the special case when the n points are in convex position, we find longer monochromatic non-crossing paths, of length... This bound cannot be improvved..