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Yair Hayut

Publications and source records attributed to Yair Hayut.

At least 19 recordsLinked to original sources
arXiv2026

Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem

We continue the study of compactness phenomena between the set-theoretic universe and $\mathrm{HOD}$ initiated by Goldberg--Poveda \cite{GolPov}. We focus on compactness phenomena around the power-set functions of $V$ and $\mathrm{HOD}$. We prove: (1) A singular strong limit cardinal with uncountable cofinality cannot be the first place where $\mathcal{P}(\cdot )$ and $ \mathcal{P}^{\mathrm{HOD}}(\cdot)$ disagree. (2) Assuming the existence of a measurable cardinal, $\aleph_ω$ can be the first place where $\mathcal{P}(\aleph_ω)\neq \mathcal{P}^{\mathrm{HOD}}(\aleph_ω)$, answering a question of Hayut. (3) If $κ$ is strong limit singular of uncountable cofinality, $\mathrm{HOD}$ is correct about cardinals less than or equal to $κ^+$ and the GCH holds in $\mathrm{HOD}$ below $κ^+$ then $(\mathrm{HOD}, V)$ has the $\mathrm{cf}(κ)^+$-cover property. We also show that the GCH assumption in (3) is necessary, which demonstrates that Magidor's classical Covering Theorem is optimal.

math.LO↗
arXiv2026

More notions of forcing add a square

Foreman and Magidor showed that the continuum hypothesis implies the existence of a countably-closed $\aleph_2$-cc forcing notion $\mathbb P$ for adding $\square_{\aleph_1}$. Here, we show that $\mathbb P$ may consistently be realized as an $\aleph_2$-Souslin tree. More generally, we prove that $\square_λ$ may be added by a $λ^+$-Souslin tree, providing the first analog of the Foreman--Magidor forcing at the level of successors of singular cardinals. Our construction is uniform and extends to inaccessible cardinals as well.

math.LO↗
arXiv2026

On a problem of Erdos and Hajnal

We address a question of Erdős and Hajnal about the ordinary partition relation $\aleph_{ω+1}\nrightarrow(\aleph_{ω+1},(3)_{\aleph_0})^2$. For $θ=\mathrm{cf}(λ)<λ$, assuming $2^λ=λ^+$ they proved the negative relation $λ^+\nrightarrow(λ^+,(3)_θ)^2$ and asked whether the (local instance of) GCH is indispensable. We show that this negative relation is consistent with $λ$ being a strong limit and $2^λ>λ^+$. The result can be pushed down to $\aleph_ω$.

math.LO↗
arXiv2026

Isomorphism Classes of Generating Sets

We introduce a new class of ultrafilters which generalizes the well-known class of simple $P$-point ultrafilters. We prove that for any well-founded $σ$-directed partial order $\mathbb{D}$ there is a mild forcing extension where there is an ultrafilter $U$ on $ω$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. On a measurable cardinal we prove a similar result: relative to a supercompact cardinal, it is consistent that $κ$ is supercompact, and for a $κ^+$-directed well-founded poset $\mathbb{D}$, there is a ${<}κ$-directed closed $κ^+$-cc forcing extension where there is a \emph{normal} ultrafilter $U$ on $κ$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. These are optimal results in the class of $P$-points and realize every potential structure of a $P$-point. We apply our constructions to obtain ultrafilters with controlled Tukey-type, in particular, an ultrafilter with non-convex Tukey and depth spectra is presented, answering questions from \cite{Benhamou_2024}. Our construction also provides new models where $\mathfrak{u}_κ<2^κ$, answering questions from \cite{Benhamou_Goldberg2025}.

math.LO↗
arXiv2026

The Gluing Property

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $κ$ and a cardinal $λ$, we say that $κ$ has the $λ$-gluing property if every sequence of $λ$-many $κ$-complete ultrafilters on $κ$ can be glued into a $κ$-complete extender. We show that every $κ$-compact cardinal has the $2^κ$-gluing property, yet non-necessarily the $(2^κ)^+$-gluing property. Finally, we compute the exact consistency-strength for $κ$ to have the $ω$-gluing property; this being $o(κ)=ω_1$.

math.LO↗
arXiv2026

Small measurable cardinals

We continue the work from [8] and make a small -- but significant -- improvement to the definition of $j$-decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extensions. In particular, this allows us to more easily lift weakly compact embeddings and thus preserve the notion of weakly critical cardinals. We use this improved lifting criterion to show that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal, both relative to the existence of a single measurable cardinal. However, if the first inaccessible cardinal is the first measurable cardinal, then in a suitable inner model it has Mitchell order of at least $2$.

math.LO↗
arXiv2026

The directedness of the Rudin-Keisler order at measurable cardinals

The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $λ\leq κ$, the following theories are equiconsistent modulo ZFC: (1) $κ$ is a measurable cardinal with $o(κ)=λ^+$ (resp. $o(κ)=κ$). (2) The Rudin-Keisler order restricted to the set of $κ$-complete (non-principal) ultrafilters on $κ$ is $λ^+$-directed (resp. $κ^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $λ$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\aleph_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be $\aleph_1$-directed.

math.LO↗
arXiv2025

The tree property on long intervals of regular cardinals

In this paper we prove that the tree property can hold on regular cardinals in an interval which overlaps a strong limit cardinal. This is a crucial milestone in the long term project, tracing back to a question raised by Foreman and Magidor in the 1980s, of obtaining the tree property at every regular cardinal above the first uncountable cardinal.

math.LO↗
arXiv2024

The first measurable can be the first inaccessible cardinal

In [8] the second and third authors showed that if the least inaccessible cardinal is the least measurable cardinal, then there is an inner model with $o(κ)\geq2$. In this paper we improve this to $o(κ)\geqκ+1$ and show that if $κ$ is a $κ^{++}$-supercompact cardinal, then there is a symmetric extension in which it is the least inaccessible and the least measurable cardinal.

math.LO↗
arXiv2024

Dense ideals

In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a $σ$-complete, $\aleph_1$-dense ideal on $\aleph_{n+1}$ for every $n < ω$, answering a question of Foreman. Using this construction we show the consistency of the existence of various irregular ultrafilters on $ω_n$, the consistency of the Foreman-Laver reflection property for the chromatic number of graphs for all possible pairs of cardinals below $\aleph_ω$, and the simultaneous consistency of the partition hypotheses $\mathrm{PH}_n(ω_m)$ for $n < m$.

math.LO↗
arXiv2024

Intermediate models with deep failure of choice

The following question was asked by Grigorieff: Suppose $V$ is a ZFC model and $V[G]$ is a set-generic extension of $V$. Can there be a ZF model $N$ so that $V\subset N \subset V[G]$ yet $N$ is not equal to $V(A)$ for any set $A\in V[G]$? The first such model was constructed by Karagila. This is the so-called \emph{Bristol model}, an intermediate model between $L$ and $L[c]$ where $c$ is a Cohen-generic real over $L$. Karagila further proves that the Kinna-Wager degree is unbounded in this model. We prove that such an intermediate extension can be found in a Cohen-generic extension of \emph{any} ground model, fully resolving Grigorieff's question. That is, let $V$ be \emph{any} ZF model and $c$ a Cohen-generic real over $V$. We prove that there is an intermediate ZF-model $V\subset N \subset V[c]$ so that $N$ is not equal to $V(A)$ for any set $A\in V[c]$, the Kinna-Wagner degree of $N$ is unbounded and, in particular, no set forcing in $N$ forces the axiom of choice. Therefore, there are class many different intermediate models of ZF between $V$ and $V[c]$.

math.LO↗
arXiv2023

Sealed Kurepa Trees

In this paper we investigate the problem of the distributivity of Kurepa trees. We show that it is consistent that there are Kurepa trees and for every Kurepa tree there is a small forcing notion which adds a branch to it without collapsing cardinals. On the other hand, we derive a proper forcing notion for making an arbitrary Kurepa tree into a non-distributive tree without collapsing $\aleph_1$ and $\aleph_2$.

math.LO↗
arXiv2023

Stationary Reflection and the failure of SCH

In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $ν$ such that the singular cardinal hypothesis fails at $ν$ and every collection of fewer than $\mathrm{cf}(ν)$ stationary subsets of $ν^+$ reflects simultaneously. For $\mathrm{cf}(ν) > ω$, this situation was not previously known to be consistent. Using different methods, we reduce the upper bound on the consistency strength of this situation for $\mathrm{cf}(ν) = ω$ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.

math.LO↗
arXiv2023

Prikry type forcings and the Bukovský-Dehornoy phenomena

This paper is meant to present in a coherent way several instances of quite common phenomena that was first identified (independently) by Bukovský and Dehornoy. We present the basic result for Prikry type forcing and show how to extend it to the Gitik-Shraon forcing, the Extender Based Prikry forcing, Prikry forcings with interleaved collapses and Radin forcing for $o(κ) < κ^+$.

math.LO↗
arXiv2023

The Spectra of transitive models

In this paper we study the spectrum of heights of transitive models of theories extending $V = L[A]$, under various definitions. In particular, we investigate the consistency strength of making those spectra as simple as possible.

math.LO↗
arXiv2023

On $ω$-Strongly Measurable Cardinals

We prove several consistency results concerning the notion of $ω$-strongly measurable cardinal in HOD. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(κ) = κ$, that every successor of a regular cardinal is $ω$-strongly measurable in HOD.

math.LO↗
arXiv2022

Complete $SE(3)$ invariants for a comeagre set of $C^3$ compact orientable surfaces in $\mathbb{R}^3$

We introduce invariants for compact $C^1$-orientable surfaces (with boundary) in $\mathbb{R}^3$ up to rigid transformations. Our invariants are certain degree four polynomials in the moments of the delta function of the surface. We give an effective and numerically stable inversion algorithm for retrieving the surface from the invariants, which works on a comeagre subset of $C^3$-surfaces.

math.DG↗