Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Trivial automorphisms

We prove that the statement `For all Borel ideals I and J on $\omega$, every isomorphism between Boolean algebras $P(\omega)/I$ and $P(\omega)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(\omega)/I$ and any other quotient $P(\omega)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $\omega$. We can also assure that the dominating number is equal to $\aleph_1$ and that $2^{\aleph_1}>2^{\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(\omega)/Fin$ are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of suslin proper forcings.Comment: Thoroughly revised versio

Tiltan

We prove that tiltan is consistent with the negation of Galvin's property. On the other hand, superclub implies Galvin's property. We also show that tiltan is consistent with a large value of the splitting number at kappa, where kappa is supercompact

Definable Combinatorics of Graphs and Equivalence Relations

Let D = (X, D) be a Borel directed graph on a standard Borel space X and let χB(D) be its Borel chromatic number. If F0, …, Fn-1: X → X are Borel functions, let DF0, …, Fn-1 be the directed graph that they generate. It is an open problem if χB(DF0, …, Fn-1) ∈ {1, …, 2n + 1, ℵ0}. Palamourdas verified the foregoing for commuting functions with no fixed points. We show here that for commuting functions with the property that there is a path from each x ∈ X to a fixed point of some Fj, there exists an increasing filtration X = ⋃m &lt; ω Xm such that χB(DF0, …, Fn-1↾ Xm) ≤ 2n for each m. We also prove that if n = 2 in the previous case, then χB(DF0, F1) ≤ 4. It follows that the approximate measure chromatic number χapM(D) ≤ 2n + 1 when the functions commute. If X is a set, E is an equivalence relation on X, and n ∈ ω, then define [X]nE = {(x0, ..., xn - 1) ∈ nX: (∀i,j)(i ≠ j → ¬(xi E xj))}. For n ∈ ω, a set X has the n-Jónsson property if and only if for every function f: [X]n= → X, there exists some Y ⊆ X with X and Y in bijection so that f[[Y]n=] ≠ X. A set X has the Jónsson property if and only for every function f : (⋃n ∈ ω [X]n=) → X, there exists some Y ⊆ X with X and Y in bijection so that f[⋃n ∈ ω [Y]n=] ≠ X. Let n ∈ ω, X be a Polish space, and E be an equivalence relation on X. E has the n-Mycielski property if and only if for all comeager C ⊆ nX, there is some Borel A ⊆ X so that E ≤B E ↾ A and [A]nE ⊆ C. The following equivalence relations will be considered: E0 is defined on ω2 by x E0 y if and only if (∃n)(∀k &gt; n)(x(k) = y(k)). E1 is defined on ω(ω2) by x E1 y if and only if (∃n)(∀k &gt; n)(x(k) = y(k)). E2 is defined on ω2 by x E2 y if and only if ∑{1⁄(n + 1): x(n) ≠ y(n)} &lt; ∞. E3 is defined on ω(ω2) by x E3 y if and only if (∀n)(x(n) E0 y(n)). Holshouser and Jackson have shown that ℝ is Jónsson under AD. The present research will show that E0 does not have the 3-Mycielski property and that E1, E2, and E3 do not have the 2-Mycielski property. Under ZF + AD, ω2/E0 does not have the 3-Jónsson property. Let G = (X, G) be a graph and define for b ≥ 1 its b-fold chromatic number χ(b)(G) as the minimum size of Y such that there is a function c from X into b-sets of Y with c(x) ∩ c(y) = ∅ if x G y. Then its fractional chromatic number is χf(G) = infb χ(b)(G)⁄b if the quotients are finite. If X is Polish and G is a Borel graph, we can also define its fractional Borel chromatic number χfB(G) by restricting to only Borel functions. We similarly define this for Baire measurable and μ-measurable functions for a Borel measure μ. We show that for each countable graph G, one may construct an acyclic Borel graph G' on a Polish space such that χfBM(G') = χf(G) and χBM(G') = χ(G), and similarly for χfμ and χμ. We also prove that the implication χf(G) = 2 ⇒ χ(G) = 2 is false in the Borel setting.</p

Measures, slaloms, and forcing axioms

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