Laver Forcing

• Laver Forcing is a set-theoretic forcing notion using trees with unique stems and infinitely many successors to add a generic real while preserving key invariants.
• It distinguishes between adding Cohen and half–Cohen reals through sophisticated Katětov order analyses, ensuring no random reals are introduced.
• Its applications extend to descriptive set theory and generalized Baire spaces, refining cardinal invariants and enabling canonization of equivalence relations.

Laver forcing is a central tool in set-theoretic forcing, notable for its deep connections to domination, combinatorics on ideals, preservation properties, and the fine structure of definable equivalence relations. Variants parameterized by co-ideals and generalizations to uncountable contexts reveal sharp distinctions between the addition of reals of various combinatorial types and the preservation of classical set-theoretic invariants.

1. Definitions: Classical Laver Forcing and Co-Ideal Variants

Classical Laver forcing $\mathbb{L}$ consists of trees $T \subseteq \omega^{<\omega}$ with a unique finite stem $\operatorname{stem}(T)$ such that every node $\tau \in T$ extending the stem has infinitely many immediate successors. The partial order is inclusion: $S \leq T$ iff $S \subseteq T$. Forcing with $\mathbb{L}$ adds a generic real $r_{\text{gen}} \in \omega^{\omega}$ as the unique path through the intersection of the generic filter (Rosa et al., 14 Jan 2026).

Generalizing, for an ideal $I$ on $\omega$, define $I^+ = \{ X \subseteq \omega : X \notin I \}$. Laver forcing associated to $I$, denoted $\mathbb{L}(I^+)$, uses the same setup but replaces "infinitely many successors" at each node $\tau$ with "successors in $I^+$". Thus, every $\tau \geq \operatorname{stem}(T)$ satisfies $\operatorname{succ}_T(\tau) \in I^+$. When $I = \operatorname{Fin}$, one recovers the classical $\mathbb{L}$.

2. Addition of Cohen and Random Reals: Katětov Order and Characterizations

The addition of Cohen reals by Laver-like forcings is captured via the Katětov order between ideals. For ideals $I, J$ on $\omega$, $I \leq_K J$ if there is $f: \omega \to \omega$ such that for all $A \in I$, $f^{-1}[A] \in J$.

A fundamental theorem states: $\mathbb{L}(I^+)$ adds a Cohen real if and only if there exists $X \in I^+$ such that the nowhere dense ideal $\operatorname{nwd}$ satisfies $\operatorname{nwd} \leq_K I \upharpoonright X$, where $$I \upharpoonright X = \{ A \subseteq X : A \in I \}$$. The proof constructs, via such a Katětov reduction, a name that is forced to be Cohen, or else uses new reals in the extension to build a witness for the order (Rosa et al., 14 Jan 2026).

In contrast, no Laver forcing (classical or co-ideal) adds random reals. This follows from the "pure decision property" and the inability to add bounded eventually different reals: any name $\dot{r} \leq_\infty f$ in the extension is forced to agree infinitely often with some ground model $g$, and thus cannot be random (Rosa et al., 14 Jan 2026).

3. The Laver Property and its Independence from Cohen Real Addition

A forcing $\mathbb{P}$ has the Laver property if every $\mathbb{P}$-name $\dot{f}$ for a function in $\omega^{\omega}$, bounded by a ground model function, is eventually captured by a ground model slalom of controlled width. The Laver property is strictly stronger than the statement that no Cohen reals are added: there exist F$_\sigma$ ideals $I$, even those extendable to P-point ultrafilters $U$, with $\operatorname{nwd} \not\leq_K I$ (so $\mathbb{L}(I^+)$ does not add Cohen reals), yet $\mathscr{L}_f \leq_K I$ holds (so $\mathbb{L}(I^+)$ fails the Laver property). Here, $\mathscr{L}_f$ is a prototype ideal generated by slaloms $Z$ of width $n+1$, controlling branching analogously to slalom bounding (Rosa et al., 14 Jan 2026).

This separation demonstrates that for co-ideal Laver forcings, preventing Cohen real addition does not guarantee the full Laver property, even in ultrafilter cases.

4. Forcing Infinitely-Often Equal (Half–Cohen) Reals

A real $h \in \omega^{\omega}$ is infinitely-often equal over $V$ if, for every ground model $f \in \omega^{\omega}$, $|\{ n : h(n) = f(n) \}| = \aleph_0$. The construction of forcings that add half–Cohen but not full Cohen reals is delicate. Define an ideal $HC$ on $\omega^{<\omega}$ by $HC$ generated from $H_r = \{ s : s \cap r = \emptyset \}$ for $r \in \omega^\omega$.

If $I$ and $X \in I^+$ satisfy $HC \leq_K I \upharpoonright X$, then both $\mathbb{L}(I^+)$ and certain tree-like forcings add an infinitely-often equal (half–Cohen) real (Rosa et al., 14 Jan 2026). However, the Katětov obstruction for $\operatorname{nwd}$ reappears on some $HC$-positive trees (notably Goldstern–Shelah trees), leading to inevitable Cohen addition along some branches. One can find antichains $X \in HC^+$ avoiding this, but these do not yield systemic tree-like forcings. The existence of a combinatorial tree-like forcing that adds exactly a half–Cohen, not Cohen, real remains unresolved (Rosa et al., 14 Jan 2026).

5. Canonization and Equivalence Relations on Laver Trees

A major application of Laver forcing arises in descriptive set theory via canonization for equivalence relations. Doucha proved a Silver-type dichotomy for Borel equivalence relations $E \leq_B E_I$ on $[T]$ for $T$ a Laver tree and $I$ an F$_\sigma$ P-ideal. For every such $E$, some subtree $S \leq T$ witnesses $E$ collapses to the identity or universal relation on $[S]$ (Doucha, 2012).

This demonstrates that the Laver ideal $\mathcal{I}_L$—generated by sets of reals vulnerable to being infinitely outside explicit slaloms—has powerful canonization properties, positioning Laver forcing within the field of canonical Ramsey theory for Polish spaces.

6. Generalizations: Laver Forcing in Generalized Baire Space

Attempts to generalize Laver forcing to uncountable regular cardinals $\kappa$ yield divergent behavior. In the generalized Baire space $\kappa^{\kappa}$, a $\kappa$–Laver tree is a limit-closed $T \subseteq \kappa^{<\kappa}$, with a stem $s$ such that every $\tau \geq s$ splits into exactly $\kappa$ immediate successors.

Any $\kappa$-Laver-type forcing (closed under subtree pruning) necessarily adds a Cohen $\kappa$-real, regardless of the details of the splitting requirement. Even further, for $<\kappa$-distributive tree forcings on $\kappa^\kappa$ that add a dominating $\kappa$-real via a ground-model continuous function, Cohen $\kappa$-reals are unavoidably added. Thus, the "Laver property" of preserving non-Cohen-ness fails spectacularly in this context, and the dichotomy results for analytic sets do not generalize (Khomskii et al., 2020).

7. Additional Structural Properties: Canonical Properties and Cardinal Invariants

Laver forcing and its co-ideal variants exhibit strong combinatorial and regularity features. For $$\mathbb{L}((\operatorname{Fin} \times \operatorname{Fin})^+)$$, the "1-1 or constant" property holds: every continuous $f: [T] \to 2^\omega$ on a Laver tree can be canonized on a subtree to have $f$ either constant or injective (Rosa et al., 14 Jan 2026).

Various cardinal invariants for ideals are computed using Laver forcings, synthesizing classic notions like the additivity of the meager ideal, covering number for meager sets, or the dominating number $\mathfrak{d}$. For instance, $$\operatorname{add}_\omega^*(\operatorname{nwd}) = \operatorname{add}(\mathcal{M})$$, and $$\operatorname{non}_\omega^*(\operatorname{Fin} \times \operatorname{Fin}) = \mathfrak{d}$$, resolving previous open questions about the interaction of these invariants with ideal-based forcings (Rosa et al., 14 Jan 2026).