According to the Erdős-Szekeres theorem, every set of n points in the plane contains roughly logn in convex position. We investigate how this bound changes if our point set does not contain a subset that belongs to a fixed order type.

An explicit coloring of the edges of Kn is constructed such that every copy of K4 has at least four colors on its edges. As n →∞, the number of colors used is n. This improves upon the previous bound of O(n) due to Erdős and Gyárfás obtained by probabilistic methods. The exponent 1/2 is optimal, since it is known that at least Ω(n) colors are required in such a coloring. The coloring is related...

Let R be a subring of the complex numbers and a be a cardinal. A system L of linear homogeneous equations with coefficients in R is called a-regular over R if, for every a-coloring of the nonzero elements of R, there is a monochromatic solution to L in distinct variables. In 1943, Rado classified those finite systems of linear homogeneous equations that are a-regular over R for all positive int...

Let L/K be a Galois extension of number fields. The problem of counting the number of prime ideals p of K with fixed Frobenius class in Gal(L/K) and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotarëv Density Theorem for this problem is small “on average.” The result may be viewed as a variation on the classical Barban-Dave...

Let ∆s = R(K3,Ks) − R(K3,Ks−1), where R(G,H) is the Ramsey number of graphs G and H defined as the smallest n such that any edge coloring of Kn with two colors contains G in the first color or H in the second color. In 1980, Erdős and Sós posed some questions about the growth of ∆s. The best known concrete bounds on ∆s are 3 ≤ ∆s ≤ s, and they have not been improved since the stating of the pro...

An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph H , the rainbow Turán number ex(n,H) is defined as themaximumnumber of edges in a properly edge-colored graph on n vertices with no rainbow copy of H . We study the rainbow Turán number of even cycles, and prove that for every fixed ε > 0, there is a constant C(ε) such that every properly edge-...

A set theoretic approach for the spectral characterization of a color scanner is described. These devices usually employ three channels to obtain a device dependent RGB (red, green, blue) image. To display/print an image, the device dependent RGB values must be correctly transformed to the color space of the target device. To determine accurate and efficient transformations for a number of devi...

In [3, 4] Törnquist and Weiss proved many natural Σ2 definable counterparts of classical equivalences to the Continuum Hypothesis (CH). These become equivalent to “all reals are constructible”. Following this scheme, we proved definable counterparts for some algebraic equivalent form of CH. More specifically we obtained a Σ2 version of a result about avoidable polynomials proven by Schmerl [2]....

A sequence S is called anagram-free if it contains no consecutive symbols r1r2 . . . rkrk+1 . . . r2k such that rk+1 . . . r2k is a permutation of the block r1r2 . . . rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Ha luszczak and Riordan [2], we consider a natural generalisation of ana...

Eighteen-month-olds successfully imitate actions, but little research has investigated their ability to imitate and generalize strategies. We explored young children’s ability to learn a sorting strategy from a model’s demonstration. In a Modeling condition (N = 24) children witnessed a model sort objects by kind. They were then given a chance to interact with the same objects, followed by new ...