Gyula Károlyi

Ramsey-type results for geometric graphs, II

Abstract. We show that for any two-coloring of the

Problems and Results around the Erdös-Szekeres Convex Polygon Theorem

{n \choose 2}

Transversals of additive Latin squares

segments determined by n points in the plane, one of the color classes contains noncrossing cycles of lengths

Chromatic variants of the Erdös-Szekeres theorem on points in convex position

3,4,\ldots,\lfloor\sqrt{n/2}\rfloor

A Turán-type Extremal Theory of Convex Geometric Graphs

. This result is tight up to a multiplicative constant. Under the same assumptions, we also prove that there is a noncrossing path of length Ω(n2/3) , 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 noncrossing paths, of length

A compactness argument in the additive theory and the polynomial method

\lfloor({n+1})/{2}\rfloor

The Erdős-Heilbronn problem in Abelian groups

. This bound cannot be improved. We also discuss some related problems and generalizations. In particular, we give sharp estimates for the largest number of disjoint monochromatic triangles that can always be selected from our segments.

Ramsey-remainder for convex sets and the Erdös-Szekeres theorem

Eszter Klein’s theorem claims that among any 5 points in the plane, no three collinear, there is the vertex set of a convex quadrilateral.An application of Ramsey’s theorem then yields the classical Erdos-Szekeres theorem [19]: For every integer n ≥ 3 there is an N0 such that, among any set of N ≥ N 0 points in general position in the plane, there is the vertex set of a convex n-gon. Let f(n) denote the smallest such number.

On Point Covers of Multiple Intervals and Axis-Parallel Rectangles

AbstractLetA={a1, …,ak} andB={b1, …,bk} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈Sksuch that the sums αi+bi, 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon’s result to the groups (ℤp)a and