For a family of geometric objects in the plane F = {S1, . . . , Sn}, define χ(F) as the least integer ` such that the elements of F can be colored with ` colors, in such a way that any two intersecting objects have distinct colors. When F is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most k pseudo-disks, it can be p...

Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles determined by n noncollinear points in the plane is at least b 2 c, which is attained for dn/2e and respectively bn/2c equally spaced points lying on two parallel lines. We show that this number is at least 17 38 n − O(1) ≈ 0.4473n. The best previous bound, ( √ 2 − 1)n − O(1) ≈ 0.4142n, which dates...

A coloring of the edges of Kn is constructed such that every copy of K4 has at least three colors on its edges. As n → ∞, the number of colors used is eO( √ log n ). This improves upon the previous probabilistic bound of O( √ n ) due to Erdős and Gyárfás.

Let f(n, p, q) be the minimum number of colors necessary to color the edges of Kn so that every Kp is at least q-colored. We improve current bounds on these nearly “anti-Ramsey” numbers, first studied by Erdős and Gyárfás. We show that f(n, 5, 9) ≥ 7 4 n− 3, slightly improving the bound of Axenovich. We make small improvements on bounds of Erdős and Gyárfás by showing 5 6 n+1 ≤ f(n, 4, 5) and f...

An open conjecture of Erdős states that for every positive integer k there is a (least) positive integer f(k) so that whenever a tournament has its edges colored with k colors, there exists a set S of at most f(k) vertices so that every vertex has a monochromatic path to some point in S. We consider a related question and show that for every (finite or infinite) cardinal κ > 0 there is a cardin...

In this paper, we consider the following problem due to Erdős: for each m ∈ N, is there a (least) positive integer f(m) so that every finite mcolored tournament contains an absorbent set S by monochromatic directed paths of f(m) vertices? In particular, is f(3) = 3? We prove several bounds for absorbent sets of m-colored tournaments under certain conditions on the number of colors of the arcs i...