arXiv subjects

José de Jesús Pelayo Gómez

Publications and source records attributed to José de Jesús Pelayo Gómez.

5 recordsLinked to original sources

arXiv2026

$\mathfrak{d}=ω_1$ implies $\mathfrak{a}=ω_1$

We prove in ZFC that $\mathfrak d=ω_1$ implies $\mathfrak a=ω_1$, settling a question of Roitman from the 1970s. Here $\mathfrak d$ is the least size of a family that eventually dominates every function in $ω^ω$, and $\mathfrak a$ is the least size of an infinite maximal almost disjoint (MAD) family of infinite subsets of $ω$. Consequently, $\mathfrak a\leq\mathfrak d$ whenever the continuum $\mathfrak c=2^{\aleph_0}$ is at most $ω_2$. This answers negatively Shelah's question whether $\mathfrak d<\mathfrak a$ is consistent with $\mathfrak c=ω_2$, and shows that his model of $\mathfrak d=ω_2<\mathfrak a=\mathfrak c=ω_3$ has the least possible values of $\mathfrak d$, $\mathfrak a$ and $\mathfrak c$ among models of $\mathfrak d<\mathfrak a$. The proof also gives a preservation theorem: from any dominating family of size $ω_1$, we construct a MAD family of size $ω_1$ that remains maximal in every outer model in which the given family is still dominating. Thus, under $\mathfrak d=ω_1$, some MAD family is indestructible by every $ω^ω$-bounding forcing; it can moreover be chosen almost strongly separable, hence Cohen-indestructible. These results give negative answers, under $\mathfrak d=ω_1$, to two questions of Hrušák and to one of Brendle, Guzmán, Hrušák and Raghavan.

math.LO↗

arXiv2026

Infinite Games and Ramsey Properties of $F_σ$ Ideals

We study combinatorial properties of Borel ideals on countable sets. We construct a tall $F_σ$ ideal for which player II has a winning strategy in the cut-and-choose game, answering a question of J. Zapletal. We also generalize the characterization of Ramsey properties by the random graph ideal to ideals associated with random colorings of finite subsets. Extending a construction of Hrušák, Meza-Alcántara, Thümmel, and Uzcátegui, we obtain, for each $m\inω$, a tall $F_σ$ ideal $\mathcal{ED}_m$ satisfying $ω\to(\mathcal{ED}_m^+)^2_{m+1}$ but not $ω\to(\mathcal{ED}_m^+)^2_{m+2}$. We also construct a tall $F_{σδ}$ ideal $\mathcal S_ω$ extending the Solecki ideal, with $\mathcal S_ω^+\to(ω,\mathcal S_ω^+)^2_2$ but $\mathcal S_ω^+\not\to(\mathcal S_ω^+)^2_2$, and derive a Ramsey-type property of the Solecki ideal. Finally, we construct a tall $F_σ$ $K$-uniform ideal that is not Katětov-equivalent to $\mathcal{ED}_{\mathrm{fin}}$, answering a question of M. Hrušák.

math.LO↗

arXiv2026

The measure half of the $2n+1$ problem

Let $F_0,\dots,F_{n-1}$ be Borel functions on a standard Borel space $X$ and let $G_F$ be the graph they generate. We prove that for every Borel probability measure $μ$ on $X$ there are a forward-invariant $μ$-conull Borel set $A \subseteq X$ and a proper Borel $(2n+1)$-coloring of the restriction of $G_F$ to $A$; consequently $χ_M(G_F) \le 2n+1$ in the total sense of Kechris and Marks. This answers positively, for every $n$ and with the optimal constant, the measure half of Problem 5.14 of the survey "Descriptive graph combinatorics" of Kechris and Marks -- the measurable version of a question raised by Kechris, Solecki, and Todorcevic (Adv. Math. 141, 1999). No invariance, no local finiteness, and no finiteness of the Borel chromatic number are assumed. The argument is elementary and specific to measure; the corresponding Borel problem and the Baire-measurable half remain open.

math.LO↗

arXiv2026

Tree-derived ideals: Fubini iterations, limit amalgamations, and Katetov obstructions

We develop a machinery for deriving ideals on a countable set from a partition of $ω$ indexed by $ω^{<ω}$. A derivative operator on trace trees, parametrized by an auxiliary ideal $\mathcal J$, yields a strict transfinite hierarchy $\mathcal H^{\mathcal J}_α$ of proper ideals, tall from level one onward and independent of the chosen partition. Its finite levels are exactly the Fubini powers, $\mathcal H_n\cong\mathrm{Fin}^{\otimes(n+1)}$, while $\mathcal H_ω=\bigcup_n\mathcal H_n$ amalgamates all finite powers. We establish presentation independence, local homogeneity, Fubini recursion, and $Π^1_1$-completeness of the full hierarchy. Let $\mathcal F_ω$ be Kwela's canonical inductive limit and $\mathcal F'_ω$ his independent-partitions limit. We prove $\mathcal F_ω\not\leq_K\mathcal H_ω$, although $\mathcal F'_ω\sqsubseteq\mathcal H_ω$ and every finite coherent fragment of a putative reduction is realizable over $\mathcal H_ω$. The proof introduces essential depth, an invariant monotone along Katetov reductions of $\mathrm{Fin}\otimes\mathrm{Fin}$, and gives the sharp non-extension bound $N(m)=m+2$. Thus $\mathcal H_ω$ contains no isomorphic copy of $\mathcal F_ω$, and $\mathcal F_ω\not\leq_K\mathcal F'_ω$. Applications include chromatic ideals $\mathcal G_k\in\mathcal H_2\setminus\mathcal H_1$ whose inclusion order records divisibility and whose Katetov order records arithmetic. We also prove the orthogonality of Cantor--Bendixson and derivative ranks. Finally, $\mathcal P(ω)/\mathcal H$ is a $σ$-closed reduced power $\mathbb B\cong\mathbb B^ω/\mathrm{Fin}$, contains $\mathcal P(ω)/\mathrm{Fin}$ regularly, and under CH is forcing-equivalent to $(\mathcal P(ω)/\mathrm{Fin})^+$.

math.LO↗

arXiv2026

Reductions and necessary conditions for tall Borel Ramsey ideals

An ideal $\mathcal{I}$ on $ω$ has the Ramsey property if $\mathcal{I}^{+}\to(\mathcal{I}^{+})^2_2$: every $2$-colouring of the pairs of an $\mathcal{I}$-positive set has an $\mathcal{I}$-positive homogeneous subset. Whether a tall Borel ideal can have the Ramsey property is an open question of Hrušák, Meza-Alcántara, Thümmel and Uzcátegui; a coanalytic example exists in ZFC, so a negative answer must use definability essentially. Our main theorem, a synthesis of the results of the paper, states that a tall Borel Ramsey ideal admits no countable local reading. Below every positive set, such an ideal is not a countable intersection of topologically represented (or tall analytic $P$-) ideals, and its quotient has no countable dense subset. Moreover, every quotient name for a new real has uncountable width, the colouring witnessing non-selectivity of the generic ultrafilter is never read continuously on a positive condition, and hereditary tall subfamilies saturate every finite window of barrier dimensions coherently but never all dimensions at once. We prove separately that a weakly selective $q^+$ ideal admits no positive $\mathcal{ED}_{\mathrm{fin}}$-carrier. The converse question -- must Borelness force a properness-like countable reading on some positive condition? -- is stated in three precise forms with proved consequences: two of them would refute tall Borel Ramsey ideals outright, the third the strictly weaker Nash--Williams class. The main question remains open.

math.LO↗