The Rainbow Ramsey Theorem is essentially an “anti-Ramsey” theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey’s Theorem, even in the weak system RCA0 of reverse

Abstract. We prove induced Ramsey theorems in which the induced monochromatic subgraph satisfies that some of its partial automor-phisms extend to automorphisms of the colored graph. 1.

In certain circumstances, one-dimensional parametrized Ramsey theorems extend to infinite dimension. As a consequence, countable products of many forcings do not add splitting reals.

Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable

I prove forcing preservation theorems for products of definable partial orders preserving cofinality of the meager or null ideal. Rectangular Ramsey theorems for the related ideals follow from the proofs.

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It is known that if P and Q are posets and is lexicographic product, then (in the Erd}os-Rado partition notation), PQ ! (P; Q).

According to a classical result of Szemerédi, every dense subset of 1, 2,..., N contains an arbitrary long arithmetic progression, if N is large enough. Its analogue in higher dimensions due to Fürstenberg and Katznelson says that every dense subset of {1, 2,..., N} d contains an arbitrary large grid, if N is large enough. Here we present geometric variants of these results for separated point sets on the line and respectively in the Euclidean space: (i) every dense separated set of points in some interval [0, L] on the line contains an arbitrary long approximate arithmetic progression, if L is large enough. (ii) every dense separated set of points in the d-dimensional cube [0, L] d in R d contains an arbitrary large approximate grid, if L is large enough. A further generalization for any finite pattern in R d is also established. The separation condition is shown to be necessary for such results to hold. In the end we show that every sufficiently large point set in R d contains an arbitrarily large subset of almost collinear points. No separation condition is needed in this case.

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