Constant Evasion Number in Cardinal Characteristics

• The constant evasion number is a cardinal invariant defined through a bounded-gap prediction framework, requiring predictors to be correct within every fixed-length interval.
• It is expressed via relational systems and ideal-theoretic formulations, linking the minimal sizes of reals and predictor families to invariants like add(M) and cof(M).
• Forcing constructions embed these invariants into Cichoń-style models, clearly separating constant prediction properties from classical cardinal characteristics.

The constant evasion number is a cardinal characteristic of the continuum defined from a bounded-gap version of prediction on reals. For $n\le \omega$, a predictor is a function $\sigma:{}^{<\omega}n\to n$, and a real $f\in{}^\omega n$ is constantly predicted by $\sigma$ when there exists a fixed $k\in\omega$ such that every interval of length $k$ contains some coordinate at which the predictor is correct: $$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$ The constant evasion number $e_n^{const}$ is the least size of a family of reals that no single predictor can constantly predict all at once, while the dual constant prediction number $v_n^{const}$ is the least size of a family of predictors covering all reals under the same relation. Recent work places these invariants within the modern theory of relational systems, ideals, and forcing, and shows that the binary instances can be separated from the classical characteristics in Cichoń-style models (Cardona et al., 15 Jul 2025, Cardona et al., 31 Mar 2025).

1. Definition and bounded-gap prediction

The defining feature of constant evasion is the replacement of ordinary eventual prediction by a bounded-gap success criterion. Instead of asking that a predictor be correct at all sufficiently large stages, one asks for a finite bound $k$ such that every interval $\sigma:{}^{<\omega}n\to n$0 contains at least one correct prediction. In the notation used for the binary case,

$\sigma:{}^{<\omega}n\to n$1

means

$\sigma:{}^{<\omega}n\to n$2

so the predictor need only hit the target repeatedly with bounded gaps rather than stabilize to correctness at every later coordinate (Cardona et al., 31 Mar 2025).

For general $\sigma:{}^{<\omega}n\to n$3, the constant evasion number is

$\sigma:{}^{<\omega}n\to n$4

and the dual constant prediction number is

$\sigma:{}^{<\omega}n\to n$5

Thus $\sigma:{}^{<\omega}n\to n$6 measures how many reals are needed before every single predictor fails on at least one of them, whereas $\sigma:{}^{<\omega}n\to n$7 measures how many predictors are needed to cover the whole space (Cardona et al., 15 Jul 2025).

This formulation is a refinement of the classical prediction/evasion framework of Blass. A plausible implication is that constant evasion isolates a stronger obstruction than ordinary evasion: the predictor is not merely required to succeed infinitely often, but to do so with uniformly bounded waiting time between successes.

2. Relational-system and ideal-theoretic formulations

The constant evasion number is naturally expressed as the bounding number of a relational system. For equality on base $\sigma:{}^{<\omega}n\to n$8, the relevant system is

$\sigma:{}^{<\omega}n\to n$9

with

$f\in{}^\omega n$0

This places constant evasion directly inside the Tukey-theoretic language used for many cardinal characteristics (Cardona et al., 15 Jul 2025).

The framework extends beyond equality. Given $f\in{}^\omega n$1 with $f\in{}^\omega n$2 and a sequence of relations $f\in{}^\omega n$3, one defines

$f\in{}^\omega n$4

and from this the invariants $f\in{}^\omega n$5 and $f\in{}^\omega n$6. The notational special cases include $f\in{}^\omega n$7 for equality, $f\in{}^\omega n$8 for inequality, and $f\in{}^\omega n$9 for the order relation $\sigma$0 (Cardona et al., 15 Jul 2025).

A major structural result is the ideal representation. For $\sigma$1 and $\sigma$2,

$\sigma$3

and the associated ideals $\sigma$4 and $\sigma$5 satisfy

$\sigma$6

The paper also proves

$\sigma$7

together with

$\sigma$8

These identifications show that constant evasion is not only a combinatorial parameter on predictors, but also an ideal-theoretic invariant with additivity and cofinality interpretations (Cardona et al., 15 Jul 2025).

One especially notable consequence is

$\sigma$9

which gives a constant-prediction characterization of the meager ideal in the $k\in\omega$0-setting (Cardona et al., 15 Jul 2025).

3. Position among classical cardinal characteristics

The constant evasion number belongs to the broader prediction/evasion hierarchy. In the classical setup, a predictor is a pair

$k\in\omega$1

where $k\in\omega$2 is infinite and each $k\in\omega$3; the classical evasion number $k\in\omega$4 is the least size of a family of reals defeating every predictor, and the corresponding prediction number is $k\in\omega$5 (Yamazoe, 2024). That earlier work showed that $k\in\omega$6 can itself be inserted into Cichoń’s maximum as a distinct characteristic: $k\in\omega$7 This provides the immediate backdrop for the constant variants (Yamazoe, 2024).

For the constant framework, the general inequalities recorded in the literature include

$k\in\omega$8

$k\in\omega$9

and

$k$0

Dually,

$k$1

$k$2

and

$k$3

The same source recalls

$k$4

hence in particular $k$5 (Cardona et al., 15 Jul 2025).

In the notation of the binary forcing paper, several further comparisons are emphasized: $k$6 and the consistency statements

$k$7

These comparisons indicate that the constant invariants interact nontrivially with the null, meager, and $k$8-ideal characteristics rather than collapsing to a single pre-existing value (Cardona et al., 31 Mar 2025).

4. Forcing constructions and Cichoń-style separation

A central recent result is that the binary constant evasion and constant prediction invariants can be threaded into a full Cichoń-style configuration with pairwise distinct values. The forcing results produce ccc forcing notions realizing constellations in which the classical characteristics and the constant prediction/evasion invariants take prescribed regular values (Cardona et al., 31 Mar 2025).

One displayed configuration yields

$k$9

together with the corresponding placement of $$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$0 on the right-hand side of the diagram. A second theorem realizes

$$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$1

and a third gives

$$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$2

The point of these theorems is that the constant invariants can be inserted into the “maximum” configuration rather than merely separated in isolated models (Cardona et al., 31 Mar 2025).

The proofs combine several forcing technologies: ultrafilter-limit iterations, finitely additive measure/fam-limit iterations, simple matrix iterations, preservation of goodness for suitable relational systems, and the new combinatorial property $$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$3 together with a related $$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$4-$$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$5-cc condition. A key forcing $$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$6, for finite $$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$7, generalizes Brendle’s forcing for constant prediction and adds a generic predictor that constantly predicts all ground-model reals in $$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$8; this poset is proved to be uniformly $$f^{cp}_{=}\sigma \iff \exists k\in\omega\ \forall i\in\omega\ \exists j\in[i,i+k)\ f(j)=\sigma(f\upharpoonright j).$$9-uf-lim-linked. The paper also proves that $e_n^{const}$0 is $e_n^{const}$1-uf-linked (Cardona et al., 31 Mar 2025).

Another major ingredient is the preservation theorem for $e_n^{const}$2: if each iterand in a finite-support iteration satisfies $e_n^{const}$3, then the whole iteration does too. This is used to preserve strongly unbounded families for the relational system $e_n^{const}$4, from which the authors derive

$e_n^{const}$5

The final separation step uses the standard submodel method: for a large ccc forcing $e_n^{const}$6, one intersects with a suitably chosen $e_n^{const}$7-closed elementary submodel $e_n^{const}$8, and a preservation lemma transfers relations of the form $e_n^{const}$9 or $v_n^{const}$0 to $v_n^{const}$1 and $v_n^{const}$2 (Cardona et al., 31 Mar 2025).

5. Generalizations, dualities, and sensitivity to small changes

The constant evasion number sits inside a large family of variants. The paper on generalizations treats equality, inequality, order, varying bases $v_n^{const}$3, “ubd” and “hyp” envelopes, and the eventual-prediction analogues

$v_n^{const}$4

defined from

$v_n^{const}$5

There is a direct Tukey comparison

$v_n^{const}$6

hence

$v_n^{const}$7

For equality, the paper also shows that the version based on limited predictors collapses to the unrestricted one: $v_n^{const}$8 These results indicate that the bounded-gap relation defines a robust hierarchy rather than a single isolated invariant (Cardona et al., 15 Jul 2025).

A complementary theme is sensitivity to small changes in the underlying relation. The paper on zero-prediction modifies the classical prediction system only on the countable set

$v_n^{const}$9

producing the invariant $k$0. It proves

$k$1

and shows that for any uncountable regular cardinals $k$2, it is consistent that

$k$3

By contrast, for Blass’s global width-$k$4 prediction invariant $k$5, the same $k$6-operation leaves the cardinal unchanged: $k$7 This suggests that prediction-based cardinal invariants can be highly sensitive to how the relation is altered on a countable exceptional class, but that such sensitivity depends on the geometry of the relational system (Yamazoe, 26 May 2026).

6. Terminology, adjacent usages, and open problems

In set theory, the constant evasion number refers to the bounded-gap prediction invariant on reals described above. The phrase can be confused with unrelated uses in graph theory and geometric pursuit-evasion.

In the graph-theoretic eviction game, the relevant parameter is the eviction number $k$8, the minimum size of the sets in an eternal dominating family. That literature proves that bounded independence number implies bounded eviction number via a Ramsey-based function $k$9, but it does not prove a universal constant-factor estimate

$\sigma:{}^{<\omega}n\to n$00

Indeed, the existence of such a constant $\sigma:{}^{<\omega}n\to n$01 is posed as an open problem (MacGillivray et al., 23 Sep 2025).

In complete-information pursuit-evasion in polygonal environments, the phrase “constant evasion number” refers to a constant number of pursuers needed for capture: three pursuers are always sufficient and sometimes necessary, independently of the number of vertices or holes (Klein et al., 2011). This is conceptually unrelated to the set-theoretic cardinal characteristic.

Within set theory itself, several open directions remain. The generalization paper ends with questions including

$\sigma:{}^{<\omega}n\to n$02

together with the corresponding dual problems (Cardona et al., 15 Jul 2025). The zero-prediction paper asks whether ZFC proves

$\sigma:{}^{<\omega}n\to n$03

equivalently whether $\sigma:{}^{<\omega}n\to n$04 must always hold (Yamazoe, 26 May 2026).

The current state of the subject therefore presents the constant evasion number as a genuine cardinal characteristic with multiple realizations: it has a bounded-gap combinatorial definition, a relational-system formulation, an ideal-theoretic interpretation, a rich forcing theory, and a developing network of variants whose behavior is only partly understood.