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Register a new userInfinite monochromatic sumsets for colourings of the reals
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N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $mathbb R$ so that no infinite sumset $X+X={x+y:x,yin X}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:mathbb Rto r$ with $r$ finite there is an infinite $Xsubseteq mathbb R$ so that $c$ is constant on $X+X$.
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A traversal of a connected graph is a linear ordering of its vertices all of whose initial segments induce connected subgraphs. Traversals, and their refinements such as breadth-first and depth-first traversals, are computed by various graph searching algorithms. We extend the theory of generic search and breadth-first search from finite graphs to wellordered infinite graphs, recovering the notion of search trees in this context. We also prove tight upper bounds on the extent to which graph search and breadth-first search can modify the order type of the original graph, as well as characterize the traversals computed by these algorithms as lexicographically minimal.
Let $M^sharp_n(mathbb{R})$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n(mathbb{R})$ the class-sized model obtained by iterating the topmost measure of $M_n(mathbb{R})$ class-many times. We characterize the sets of reals which are $Sigma_1$-definable from $mathbb{R}$ over $M_n(mathbb{R})$, under the assumption that projective games on reals are determined: (1) for even $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Pi^1_{n+1}$; (2) for odd $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Sigma^1_{n+1}$. This generalizes a theorem of Martin and Steel for $L(mathbb{R})$, i.e., the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $mathbb{R}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists for all $ninmathbb{N}$, and that determinacy of all projective games of length $omega^2$ with moves in $mathbb{N}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists and satisfies $mathsf{AD}$ for all $ninmathbb{N}$.
This article discusses some recent trends in Ramsey theory on infinite structures. Trees and their Ramsey theory have been vital to these investigations. The main ideas behind the authors recent method of trees with coding nodes are presented, showing how they can be useful both for coding structures with forbidden configurations as well as those with none. Using forcing as a tool for finite searches has allowed the development of Ramsey theory on such trees, leading to solutions for finite big Ramsey degrees of Henson graphs as well as infinite dimensional Ramsey theory of copies of the Rado graph. Possible future directions for applications of these methods are discussed.
We investigate the transfinite game values arising in infinite chess, providing both upper and lower bounds on the supremum of these values---the omega one of chess---with two senses depending on whether one considers only finite positions or also positions with infinitely many pieces. For lower bounds, we present specific infinite positions with transfinite game values of omega, omega^2, omega^2 times k, and omega^3. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true omega one.
We construct a model of $mathsf{ZF} + mathsf{DC}$ containing a Luzin set, a Sierpi{n}ski set, as well as a Burstin basis but in which there is no a well ordering of the continuum.
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