Proof. The case $\kappa =\theta ^+$ is covered by [Reference Inamdar and RinotIR23, Corollary 7.3], so suppose that we are given a $\theta \in \operatorname {\mathrm {Reg}}(\kappa )$ with $\theta ^+<\kappa $ . By Shelah’s club-guessing theorem [Reference ShelahShe94b, Claim 2.7], we may pick a sequence $\vec C=\langle C_\delta \mid \delta \in E^\kappa _\theta \rangle $ such that:

• • For every $\delta \in E^\kappa _\theta $ , $C_\delta $ is a club in $\delta $ of order-type $\theta $ ;

• • For every club $C\subseteq \kappa $ , there exists a $\delta \in E^\kappa _\theta $ such that $C_\delta \subseteq C$ .

As there is a $\kappa $ -Aronszajn tree, a standard argument (see [Reference Meir Brodsky and RinotBR17, Appendix]) yields a $\kappa $ -Aronszajn subtree of $({}^{<\kappa }2,{\subseteq })$ ,Footnote ⁵ i.e., a $T\subseteq {}^{<\kappa }2$ satisfying all of the following:

1. (i) For every $t\in T$ and $\alpha <\operatorname {\mathrm {dom}}(t)$ , $t\restriction \alpha \in T$ ;

2. (ii) For every $\alpha <\kappa $ , the set $T_\alpha :=\{ t\in T\mid \operatorname {\mathrm {dom}}(t)=\alpha \}$ satisfies $0<|T_\alpha |<\kappa $ ;

3. (iii) For every $f:\kappa \rightarrow 2$ , there exists an $\alpha <\kappa $ such that $f\restriction \alpha \notin T$ .

For every $\gamma <\kappa $ , denote $T\restriction \gamma :=\bigcup _{\alpha <\gamma }T_\alpha $ . Fix an injective enumeration $\langle a_\eta \mid \eta <\kappa \rangle $ of T. Finally, define $c:[\kappa ]^2\rightarrow \theta $ as follows. Given $\eta <\beta <\kappa $ , if $\operatorname {\mathrm {dom}}(a_\eta )\in E^\kappa _\theta $ and $a_\eta \nsubseteq a_\beta $ , then let

$$ \begin{align*}c(\eta,\beta):=\sup(\operatorname{\mathrm{otp}}(C_{\operatorname{\mathrm{dom}}(a_\eta)}\cap\Delta(a_\eta,a_\beta)));\end{align*} $$

otherwise, just let $c(\eta ,\beta ):=0$ .

To see that c witnesses $\operatorname {{\mathsf{onto}}}(J^{\text {bd}}[\kappa ],\theta )$ , let an arbitrary $B\in [\kappa ]^\kappa $ be given. Denote

$$ \begin{align*}T^{\leadsto B}:=\{ t\in T\mid |\{ \beta\in B\mid t\sqsubseteq a_\beta\}|=\kappa\}.\end{align*} $$

Let C be as given by the claim. Recalling the choice of $\vec C$ , we now fix a $\delta \in E^\kappa _\theta $ such that $C_\delta \subseteq C$ . Pick $t\in (T^{\leadsto B})_\delta $ . For every $\gamma \in C_\delta $ , as $\gamma ^+:=\min (C_\delta \setminus (\gamma +1))$ belongs to C, we may pick a node $t_\gamma \in (T^{\leadsto B})_{\gamma ^+}$ such that $t_\gamma \restriction (\gamma +1)=t\restriction (\gamma +1)$ and $t_\gamma \neq t\restriction \gamma ^+$ , so that $\gamma <\Delta (t,t_\gamma )<\gamma ^+$ . Pick $\eta <\kappa $ such that $t=a_\eta $ .

For each $\gamma \in C_\delta $ , as $t_\gamma \in T^{\leadsto B}$ , we may pick some $\beta _\gamma \in B$ above $\eta $ with $t_\gamma \sqsubseteq a_{\beta _\gamma }$ . Then $\Delta (a_\eta ,a_{\beta _\gamma })=\Delta (t,t_\gamma )$ and so recalling that $\gamma <\Delta (t,t_\gamma )<\min (C_\delta \setminus (\gamma +1))$ , we infer that

$$ \begin{align*}c(\eta,\beta)=\sup(\operatorname{\mathrm{otp}}(C_{\operatorname{\mathrm{dom}}(a_\eta)}\cap\Delta(a_\eta,a_{\beta_\gamma})))=\sup(\operatorname{\mathrm{otp}}(C_\delta\cap(\gamma+1)))=\operatorname{\mathrm{otp}}(C_\delta\cap\gamma).\end{align*} $$

Therefore, $c[\{\eta \}\circledast B]=\{\operatorname {\mathrm {otp}}(C_\delta \cap \gamma )\mid \gamma \in C_\delta \}=\theta $ .

Proof. As in the beginning of the proof of Theorem 3.1, we may assume that we are given a $\theta \in \operatorname {\mathrm {Reg}}(\kappa )$ with $\theta ^+<\kappa $ , and pick a $\theta $ -bounded club-guessing sequence $\vec C=\langle C_\delta \mid \delta \in E^\kappa _\theta \rangle $ . As the ineffable tree property fails, we may fix a subset $T\subseteq {}^{<\kappa }2$ satisfying all of the following:

1. (i) For every $t\in T$ and $\alpha <\operatorname {\mathrm {dom}}(t)$ , $t\restriction \alpha \in T$ ;

2. (ii) For every $\alpha <\kappa $ , the set $T_\alpha :=\{ t\in T\mid \operatorname {\mathrm {dom}}(t)=\alpha \}$ satisfies $0<|T_\alpha |<\kappa $ ;

3. (iii) There exists a sequence $\langle b_\beta \mid \beta <\kappa \rangle \in \prod _{\beta <\kappa }T_\beta $ such that for every stationary $S\subseteq \kappa $ , there are $(\alpha ,\beta )\in [S]^2$ with $b_\alpha \nsubseteq b_\beta $ .

For every $\gamma <\kappa $ , denote $T\restriction \gamma :=\bigcup _{\alpha <\gamma }T_\alpha $ . Fix an injective enumeration $\langle a_\eta \mid \eta <\kappa \rangle $ of T. Define $c:[\kappa ]^2\rightarrow \theta $ as follows. Given $\eta <\beta <\kappa $ , if $\operatorname {\mathrm {dom}}(a_\eta )\in E^\kappa _\theta $ and $a_\eta \nsubseteq b_\beta $ , then let

$$ \begin{align*}c(\eta,\beta):=\sup(\operatorname{\mathrm{otp}}(C_{\operatorname{\mathrm{dom}}(a_\eta)}\cap\Delta(a_\eta,b_\beta)));\end{align*} $$

otherwise, just let $c(\eta ,\beta ):=0$ .

To see that c witnesses $\operatorname {{\mathsf{onto}}}(\text {NS}_\kappa ,\theta )$ holds, let a stationary $B\subseteq \kappa $ be given. Denote

$$ \begin{align*}T^{\leadsto B}:=\{ t\in T\mid \{ \beta\in B\mid t\sqsubseteq b_\beta\}\text{ is stationary}\}.\end{align*} $$

Proof. It is clear that Clause (i) holds for $T^{\leadsto B}$ . To see that Clause (ii) holds, it suffices to show that $T^{\leadsto B}$ is normal, and this follows in a proof similar to that of Claim 3.1.1, this time using the fact that $\text {NS}_\kappa $ is a $\kappa $ -complete ideal.

Next, as seen in the proof of Claim 3.1.1, in order to find a club C as in the statement of the claim, it suffices to prove that every node $t\in T^{\leadsto B}$ admits two incompatible extensions in $T^{\leadsto B}$ . Towards a contradiction, suppose that this is not the case, as witnessed by a node t. This means that $Z:=\{ z\in T^{\leadsto B}\mid z\subseteq t\text { or }t\subseteq z \}$ is a chain. Since $T^{\leadsto B}$ is normal, it follows that for every $\beta <\kappa $ , we may let $z_\beta $ denote the unique element of $Z\cap {}^\beta 2$ . Recalling Clause (iii), we infer that $\{\beta \in B\mid b_\beta =z_\beta \}$ is nonstationary. As $t\in T^{\leadsto B}$ , altogether,

$$ \begin{align*}\{\beta\in B\mid t\sqsubseteq b_\beta\ \&\ b_\beta\neq z_\beta\}=\{\beta\in B \mid \operatorname{\mathrm{dom}}(t)\le\Delta(b_\beta,z_\beta)<\beta\}\end{align*} $$

is stationary. By Fodor’s lemma, pick $\epsilon \ge \operatorname {\mathrm {dom}}(t)$ such that $\{\beta \in B\mid \operatorname {\mathrm {dom}}(t)\le \Delta (b_\beta ,z_\beta )=\epsilon \}$ is stationary. As $|T_{\epsilon +1}|<\kappa $ , we may pick $z\in T_{\epsilon +1}$ such that

$$ \begin{align*}B':=\{\beta\in B\mid \operatorname{\mathrm{dom}}(t)\le\Delta(b_\beta,z_\beta)=\epsilon<\beta\ \&\ b_\beta\restriction(\epsilon+1)=z\}\end{align*} $$

is stationary. Then $B'$ witnesses that $z\in T^{\leadsto B}$ . Furthermore, picking a $\beta \in B'$ , we see that $\operatorname {\mathrm {dom}}(t)\le \Delta (z,z_\beta )<\epsilon +1=\operatorname {\mathrm {dom}}(z)$ , which means that $t\subseteq z$ and $z\neq z_{\epsilon +1}$ , contradicting the fact that Z is a chain.

Let C be as given by the claim. Fix a $\delta \in E^\kappa _\theta $ such that $C_\delta \subseteq C$ . Pick $t\in (T^{\leadsto B})_\delta $ . For every $\gamma \in C_\delta $ , as $\gamma ^+:=\min (C_\delta \setminus (\gamma +1))$ belongs to C, we may pick a node $t_\gamma \in (T^{\leadsto B})_{\gamma ^+}$ such that $\gamma <\Delta (t,t_\gamma )<\gamma ^+$ . Pick $\eta <\kappa $ such that $t=a_\eta $ .

For each $\gamma \in C_\delta $ , as $t_\gamma \in T^{\leadsto B}$ , we may find some $\beta _\gamma \in B$ above $\eta $ with $t_\gamma \sqsubseteq b_{\beta _\gamma }$ . Then $\Delta (a_\eta ,b_{\beta _\gamma })=\Delta (t,t_\gamma )$ and hence

$$ \begin{align*}c(\eta,\beta)=\sup(\operatorname{\mathrm{otp}}(C_{\operatorname{\mathrm{dom}}(a_\eta)}\cap\Delta(a_\eta,b_{\beta_\gamma})))=\sup(\operatorname{\mathrm{otp}}(C_\delta\cap(\gamma+1)))=\operatorname{\mathrm{otp}}(C_\delta\cap\gamma).\end{align*} $$

Therefore, $c[\{\eta \}\circledast B]=\theta $ .

Fact 3.4 (Reference Inamdar and RinotIR24, Theorem B)

Suppose that $\kappa $ is a regular uncountable cardinal admitting a stationary set that does not reflect at regulars. Then, for every $\theta \in \operatorname {\mathrm {Reg}}(\kappa )$ , $\operatorname {{\mathsf{onto}}}^{++}(\mathcal J^\kappa _\kappa ,\theta )$ holds.

Proof. $(1)\implies (2)$ : By possibly enlarging $\theta $ , we may assume it is an infinite regular cardinal. By Theorem 3.1 together with [Reference Inamdar and RinotIR24, Theorem 4.1(1)], $\operatorname {{\mathsf{onto}}}^+(\mathcal J^\kappa _\kappa ,\theta ^+)$ holds. Then, by [Reference Inamdar and RinotIR24, Corollary 4.18(4)], $\operatorname {{\mathsf{onto}}}^{++}(\mathcal J^\kappa _\kappa ,\theta )$ holds.

$(2)\implies (3)$ : This is trivial.

$(3)\implies (1)$ : By [Reference Inamdar and RinotIR23, Corollary 10.8].

Proof. $(1)\implies (2)$ : By possibly enlarging $\theta $ , we may assume it is an infinite regular cardinal. By Theorem 3.3 together with [Reference Inamdar and RinotIR23, Proposition 2.26(4)], $\operatorname {{\mathsf{onto}}}^+(\text {NS}_\kappa ,\theta ^+)$ holds. Then, by [Reference Inamdar and RinotIR24, Corollary 4.18(4)], $\operatorname {{\mathsf{onto}}}^{++}(\text {NS}_\kappa ,\theta )$ holds.

$(2)\implies (3)$ : This is trivial.

$(3)\implies (1)$ : By [Reference Inamdar and RinotIR23, Lemma 11.1].

Proof. Suppose that $\theta :=\log _2(\kappa )$ is a singular cardinal.

Fix the least cardinal $\mu $ for which there exists a cardinal $\nu \in (\mu ,\kappa )$ with $\nu ^\mu \ge \kappa $ . By the preceding claim, $\mu $ exists and is smaller than $\theta $ . Fix the least $\nu \in (\mu ,\kappa )$ such that $\nu ^\mu \ge \kappa $ . In particular, $\nu ^\nu \ge \kappa $ , so that $\nu \ge \theta $ .

By the minimality of $\nu $ , $\lambda ^\mu <\kappa $ for every $\lambda <\nu $ .

Proof. Let $\langle g_\beta \mid \beta <\kappa \rangle $ be an injective enumeration of a family of functions from $\mu $ to $\nu $ . We commence with a trivial observation.

We now turn to prove each of the clauses.

(1) Let $\theta \in \operatorname {\mathrm {Reg}}(\nu )$ . Let $\langle C_\delta \mid \delta \in E^\nu _\theta \rangle $ be some $\theta $ -bounded C-sequence. Fix a bijection $\pi :\nu \leftrightarrow \mu \times \nu $ . Finally pick $c:[\kappa ]^2\rightarrow \theta $ such that for every $\eta <\nu <\beta <\kappa $ , letting $(i,\delta ):=\pi (\eta )$ ,

$$ \begin{align*}c(\eta,\beta):=\begin{cases} \operatorname{\mathrm{otp}}(C_\delta\cap g_\beta(i)),&\text{if } \delta \in E^\nu_\theta\text{ and }g_\beta(i)<\delta;\\ 0,&\text{otherwise}. \end{cases}\end{align*} $$

As $\theta \le \nu <\operatorname {\mathrm {cf}}(\varkappa )$ , the proof of [Reference Inamdar and RinotIR24, Proposition 2.9(1)] shows that any witness to $\operatorname {{\mathsf{unbounded}}}(\{\nu \},[\kappa ]^{<\varkappa },\theta )$ moreover witnesses $\operatorname {{\mathsf{unbounded}}}^+(\{\nu \},[\kappa ]^{<\varkappa },\theta )$ . Thus, we verify that c witnesses $\operatorname {{\mathsf{unbounded}}}(\{\nu \},[\kappa ]^{<\varkappa },\theta )$ . To this end, let $B\in [\kappa ]^\varkappa $ and we shall show that there exists an $\eta <\nu $ such that $c[\{\eta \}\circledast B]$ has size $\theta $ .

Without loss of generality, $\min (B)\ge \nu $ . Using Claim 4.2.1, fix an $i<\mu $ such that $X:=\{ g_\beta (i)\mid \beta \in B\}$ has size $>\theta $ . Then fix the least $\delta <\nu $ such that $\operatorname {\mathrm {otp}}(X\cap \delta )=\theta $ . Consequently, $\delta \in E^\nu _\theta $ , $\sup (X \cap \delta ) = \delta $ , and $Y:=\{ \operatorname {\mathrm {otp}}(C_\delta \cap \xi )\mid \xi \in X\cap \delta \}$ is a subset of $\theta $ of size $\theta $ . Pick $\eta <\nu $ such that $\pi (\eta )=(i,\delta )$ . Then $c[\{\eta \}\circledast B]$ covers Y, as sought.

(2) Suppose $\theta \in \operatorname {\mathrm {Reg}}(\nu )$ is such that $\theta ^+<\nu $ . By Clause (1), $\operatorname {{\mathsf{unbounded}}}^+(\{\nu \}, [\kappa ]^{<\varkappa },\theta ^+)$ holds. As $\theta $ is regular, by [Reference Inamdar and RinotIR24, Corollary 4.12], $\operatorname {{\mathsf{projection}}}(\theta ^+,\theta ^+,\theta ,1)$ holds. Then, by [Reference Inamdar and RinotIR24, Lemma 4.17(2)], $\operatorname {{\mathsf{onto}}}^+(\{\nu \},[\kappa ]^{<\varkappa },\theta )$ holds, as well.

(3) Denote $\theta :=\operatorname {\mathrm {cf}}(\nu )$ . Suppose that $\theta <\nu <\operatorname {\mathrm {cf}}(\varkappa )$ and that $\nu ':=\max \{\nu ,\mu ^{\theta }\}$ is smaller than $\varkappa $ . Let $\langle \nu _j\mid j<\theta \rangle $ be a strictly increasing sequence of cardinals, converging to $\nu $ , and fix a bijection $\pi :\nu '\leftrightarrow \nu \times {}^{\theta }\mu $ . Use Clause (2) to fix a colouring $c:[\kappa ]^2\rightarrow \theta $ witnessing $\operatorname {{\mathsf{onto}}}^+(\{\nu \},[\kappa ]^{<\varkappa },\theta )$ . Define $d:[\kappa ]^2\rightarrow \theta $ as follows. Given $\eta <\nu '<\beta <\kappa $ , let $(\bar \eta ,f):=\pi (\eta )$ , and then let $d(\eta ,\beta ):=g_\beta (f(c(\bar \eta ,\beta )))$ .

To see that d witnesses $\operatorname {{\mathsf{unbounded}}}(\{\nu '\},[\kappa ]^{<\varkappa },\nu )$ , let $B\in [\kappa ]^\varkappa $ , and we shall find an $\eta <\nu '$ such that $c[\{\eta \}\circledast B]$ has size $\nu $ .

Without loss of generality, $\min (B)\ge \nu '$ . As c witnesses $\operatorname {{\mathsf{onto}}}^+(\{\nu \},[\kappa ]^{<\varkappa },\theta )$ , we may pick some $\bar \eta <\nu $ such that for every $j<\theta $ , $B_j:=\{\beta \in B\mid c(\bar \eta ,\beta )=j\}$ is in $([\kappa ]^{<\varkappa })^+$ . By Claim 4.2.1, for each $j<\theta $ , there exists some $i_j<\mu $ such that $X_j:=\{ g_\beta (i_j)\mid \beta \in B_j\}$ has size $>\nu _j$ . Define $f:\theta \rightarrow \mu $ via $f(j):=i_j$ . Pick an $\eta <\nu '$ such that $\pi (\eta )=(\bar \eta ,f)$ . Then, for every $j<\theta $ ,

$$ \begin{align*}d[\{\eta\}\circledast B_j]=\{ g_\beta(f(c(\bar\eta,\beta)))\mid \beta\in B_j\}=\{g_\beta(f(j))\mid \beta\in B_j\}=X^j\end{align*} $$

is a subset of $\nu $ of size $>\nu _j$ . As $\sup _{j<\theta }\nu _j=\nu $ , $d[\{\eta \}\circledast B]$ is a subset of $\nu $ of size $\nu $ , as sought.

(4) Suppose that $\mu ^{\operatorname {\mathrm {cf}}(\nu )}\le \nu <\operatorname {\mathrm {cf}}(\kappa )$ . It follows that $\operatorname {\mathrm {cf}}(\nu )<\nu <\operatorname {\mathrm {cf}}(\kappa )$ and $\mu ^{\operatorname {\mathrm {cf}}(\nu )}\le \nu <\kappa $ , so that $\operatorname {{\mathsf{unbounded}}}(\{\nu \},[\kappa ]^{<\kappa },\nu )$ holds by Clause (3). As $\nu <\operatorname {\mathrm {cf}}(\kappa )$ , it is the case that $[\kappa ]^{<\kappa }\in \mathcal J^\kappa _{\nu ^+}$ , so, by [Reference Inamdar and RinotIR24, Proposition 2.9(1)], moreover $\operatorname {{\mathsf{unbounded}}}^+(\{\nu \}, [\kappa ]^{<\kappa },\nu )$ holds. In particular, we may fix a colouring $c:[\kappa ]^2\rightarrow \nu $ witnessing $\operatorname {{\mathsf{unbounded}}}^+([\kappa ]^{<\kappa },\nu )$ . Suppose now that $2^\nu =\kappa $ , and fix a bijection $\pi :\kappa \leftrightarrow \kappa \times {}^\nu \nu $ . Define $d:[\kappa ]^2\rightarrow \nu $ as follows. Given $\eta <\beta <\kappa $ , let $(\bar \eta ,f):=\pi (\eta )$ , and then let $d(\eta ,\beta ):=f(c(\bar \eta ,\beta ))$ . To see that d witnesses $\operatorname {{\mathsf{onto}}}^+([\kappa ]^{<\kappa },\nu )$ , let $B\in [\kappa ]^\kappa $ be given. By the choice of c, we may fix $\bar \eta <\kappa $ and $X\in [\nu ]^\nu $ such that for every $\xi \in X$ , the set $B_\xi :=\{\beta \in B\setminus (\bar \eta +1)\mid c(\bar \eta ,\beta )=\xi \}$ is in $[\kappa ]^\kappa $ . Fix $f\in {}^\nu \nu $ such that $f[X]=\nu $ . Find $\eta <\kappa $ such that $\pi (\eta )=(\bar \eta ,f)$ . Then, given $\tau <\nu $ , pick $\xi \in X$ such that $f(\xi )=\tau $ , and note that $\{\beta \in B\setminus (\eta +1)\mid d(\eta ,\beta )=\tau \}$ covers $B_\xi \setminus (\eta +1)$ and hence it is in $[\kappa ]^\kappa $ .

Proof. By [Reference Inamdar and RinotIR24, Theorem 4.1(2)], it suffices to prove that $\operatorname {{\mathsf{unbounded}}}^+(J^{\text {bd}}[\kappa ],\theta )$ holds. Meanwhile, by [Reference Lambie-Hanson and RinotLHR18, Lemma 2.7], $\operatorname {\mathrm {U}}(\kappa ,\kappa ,\theta ,2)$ holds. In particular, $\operatorname {\mathrm {U}}(\kappa ,2,\operatorname {\mathrm {cf}}(\theta ),2)$ holds. Then by [Reference Inamdar and RinotIR24, Lemma 5.12(1)], $\operatorname {{\mathsf{unbounded}}}^+(J^{\text {bd}}[\kappa ],\operatorname {\mathrm {cf}}(\theta ))$ holds. So, if $\theta $ is regular, then we are done.

Hereafter, suppose that $\theta $ is singular. By Lemma 4.1, we may fix a singular cardinal $\nu $ with $\theta \leq \nu <\kappa $ such that $\lambda ^{\operatorname {\mathrm {cf}}(\nu )}<\kappa \le \nu ^{\operatorname {\mathrm {cf}}(\nu )}$ for every $\lambda <\nu $ .

$\blacktriangleright $ If $\theta <\nu $ , then appealing to Lemma 4.2(2) with $\mu :=\operatorname {\mathrm {cf}}(\nu )$ , $\varkappa :=\kappa $ and the successor of $\theta $ , we get that $\operatorname {{\mathsf{onto}}}^+(\{\nu \},[\kappa ]^{<\kappa },\theta ^+)$ holds. In particular, $\operatorname {{\mathsf{onto}}}^+(J^{\text {bd}}[\kappa ],\theta )$ holds.

$\blacktriangleright $ If $\theta =\nu $ , then appealing to Lemma 4.2(3) with $\mu :=\operatorname {\mathrm {cf}}(\nu )$ and $\varkappa :=\kappa $ , we get that $\operatorname {{\mathsf{unbounded}}}(\{\nu '\},[\kappa ]^{<\kappa },\nu )$ holds for some cardinal $\nu '<\kappa $ . In particular, by [Reference Inamdar and RinotIR24, Proposition 2.9(1)], $\operatorname {{\mathsf{unbounded}}}^+(J^{\text {bd}}[\kappa ],\theta )$ holds.

Proof. Let $A\in [\kappa ]^\kappa $ , $I\in \mathcal I^\kappa _\theta $ and $B\in I^+$ be given. For all $\eta <\kappa $ and $\tau <\theta $ , denote $B^{\eta ,\ge \tau }:=\{ \beta \in B\mid c(\eta ,\beta )\ge \tau \}$ .

Proof. Otherwise, we may find a function $g:A\rightarrow \theta $ such that $B^{\eta ,{\geq } g(\eta )}\in I$ for every $\eta \in A$ . Fix $\tau <\theta $ for which the following set is cofinal in $\kappa $ :

$$ \begin{align*}A':=\{\eta\in A\mid g(\eta)=\tau\}.\end{align*} $$

As c witnesses $\operatorname {\mathrm {U}}(\kappa ,2,\theta ,2)$ , we may pick $(\eta _0,\eta _1)\in [A']^2$ such that $c(\eta _0,\eta _1)>\tau $ . As $B\in I^+$ and $B^{\eta _0,{\geq } g(\eta _0)}\cup B^{\eta _1,{\geq } g(\eta _1)}\in I$ , we may pick a $\beta \in B\setminus B^{\eta _0,{\geq } g(\eta _0)}\cup B^{\eta _1,{\geq } g(\eta _1)}$ above $\eta _1$ . As $\eta _0<\eta _1<\beta $ and c is subadditive of the second kind,

$$ \begin{align*}\tau<c(\eta_0,\eta_1)\le\max\{c(\eta_0,\beta),c(\eta_1,\beta)\}<\max\{g(\eta_0),g(\eta_1)\}=\tau.\end{align*} $$

This is a contradiction.

Let $\eta $ be given by the preceding claim, and then define a function $f:\kappa \rightarrow \theta $ via $f(\beta ):=c(\eta ,\beta )$ . By the choice of $\eta $ , for every $\tau <\theta $ , $f\restriction B^{\eta ,\ge \tau }$ is a function from an $I^+$ -set to $\theta \setminus \tau $ , so by the $\theta $ -indecomposability of I, there exists some $h(\tau )\in (\tau ,\theta )$ such that the following set is in $I^+$ :

$$ \begin{align*}B_\tau:=\{\beta\in B^{\eta,\ge\tau}\mid f(\beta)<h(\tau)\}.\end{align*} $$

Fix a sparse enough $T\in [\theta ]^\theta $ such that, for all $\tau \in T$ and $\sigma <\tau $ , $h(\sigma )<\tau $ . Then $\eta $ and T are as sought.

Proof. By Proposition 5.2 and [Reference Lambie-Hanson and RinotLHR23, Lemma 3.7].

Proof. Denote $\nu :=\mathcal C(\theta ,\theta )$ .

$\blacktriangleright $ To see that $\nu \le \mathfrak d_\theta $ , fix a sequence $\langle f_\beta \mid \beta <\mathfrak d_\theta \rangle $ of functions from $\theta $ to $\theta $ that is cofinal in $({}^\theta \theta ,{\le ^{\text {bd}}})$ . For all $\tau <\theta $ and $\beta <\mathfrak d_\theta $ , let $C_{\tau ,\beta }:=\{ \gamma \in \operatorname {\mathrm {acc}}(\theta \setminus \tau )\mid f_\beta [\gamma ]\subseteq \gamma \}$ . Now, given any club $D\subseteq \theta $ , let $\pi :\theta \rightarrow D$ denote the inverse collapsing map. Pick $\tau < \theta $ and $\beta <\mathfrak d_\theta $ such that $\pi \le ^\tau f_\beta $ . Towards a contradiction, suppose that $C_{\tau +1,\beta }\nsubseteq D$ and fix $\gamma \in C_{\tau +1,\beta }\setminus D$ . In particular, $\bar \gamma :=\sup (D\cap \gamma )$ and $\tau $ are both smaller than $\gamma $ . Put $\eta :=\max \{\bar \gamma ,\tau \}+1$ so that $\bar \gamma <\eta < \gamma $ . Then $\bar \gamma <\eta \le \pi (\eta )\le f_\beta (\eta )<\gamma $ . Altogether, $\pi (\eta )$ is an element of D lying in the interval $(\bar \gamma ,\gamma )$ . This is a contradiction.

$\blacktriangleright $ To see that $\mathfrak d_\theta \le \nu $ , fix a sequence $\langle X_\beta \mid \beta <\nu \rangle $ of cofinal subsets of $\theta $ , such that, for every club D in $\theta $ , for some $\beta <\nu $ , $X_\beta \subseteq D$ . For each $\beta <\nu $ , define $f_\beta :\theta \rightarrow X_\beta $ via $f_\beta (\eta ):=\min (X_\beta \setminus (\eta +1))$ . Now, given any function $f:\theta \rightarrow \theta $ , consider the club $D:=\{\delta <\theta \mid f[\delta ]\subseteq \delta \}$ . Fix $\beta <\nu $ such that $X_\beta \subseteq D$ , and we shall show that $f\le ^0f_\beta $ . Indeed, for every $\eta <\theta $ , $\delta :=f_\beta (\eta )$ is an element of D above $\eta $ and hence $f(\eta )<\delta =f_\beta (\eta )$ .

Proof. By Clauses (3) and (4) of Lemma 2.4 together with Lemma 5.5, in our case, implies $\operatorname {{\mathsf{onto}}}^+(\mathcal I^\kappa _\theta , \theta )$ . Thus, we may focus on proving that holds.

As $\mathfrak b_\theta =\mathfrak d_\theta =\kappa $ , we may fix a sequence $\vec g=\langle g_\beta \mid \beta <\kappa \rangle $ that is increasing and cofinal in $({}^\theta \theta ,{\le ^{\text {bd}}})$ . Define a colouring $c:[\kappa ]^2\rightarrow \theta $ by letting for all $\eta <\theta \le \beta <\kappa $ :

$$ \begin{align*}c(\eta,\beta):=g_\beta(\eta).\end{align*} $$

To see that c witnesses , let $A\in [\theta ]^\theta $ , $I\in \mathcal I^\kappa _\theta $ and $B\in I^+$ be given. For all $\eta ,\tau <\theta $ , denote $B^{\eta ,\ge \tau }:=\{ \beta \in B\mid g_\beta (\eta )\ge \tau \}$ .

Proof. Otherwise, we may define a strictly increasing function $g:\theta \rightarrow \theta $ such that $B^{\eta ,{\geq } g(\eta )}\in I$ for every $\eta \in A$ . Pick $\alpha <\kappa $ such that $g\le ^{\text {bd}} g_\alpha $ . Since $\vec g$ is increasing, it follows that we may define a function $f:B\setminus \alpha \rightarrow \theta $ via

$$ \begin{align*}f(\beta):=\min\{\eta<\theta\mid g\le^\eta g_\beta\}.\end{align*} $$

As I is $\theta $ -indecomposable, we may now fix a large enough $\eta \in A$ such that the following set is in $I^+$ :

$$ \begin{align*}B':=\{\beta\in B\setminus\alpha\mid f(\beta)\le\eta\}.\end{align*} $$

For every $\beta \in B'$ , since $f(\beta )\le \eta $ , it is the case that $g\le ^\eta g_\beta $ , in particular, $g_\beta (\eta )\ge g(\eta )$ , and hence $B^{\eta ,{\geq } g(\eta )}$ contains the positive set $B'$ . This is a contradiction.

Let $\eta $ be given by the preceding claim, and then define a function $f:\kappa \rightarrow \theta $ via $f(\beta ):=g_\beta (\eta )$ . By the choice of $\eta $ , for every $\tau <\theta $ , $f\restriction B^{\eta ,\ge \tau }$ is a function from an $I^+$ -set to $\theta \setminus \tau $ , so by the $\theta $ -indecomposability of I, there exists some $h(\tau )\in (\tau ,\theta )$ such that the following set is in $I^+$ :

$$ \begin{align*}B_\tau:=\{\beta\in B^{\eta,\ge\tau}\mid f(\beta)<h(\tau)\}.\end{align*} $$

Fix a sparse enough $T\in [\theta ]^\theta $ such that, for all $\tau \in T$ and $\sigma <\tau $ , $h(\sigma )<\tau $ . Then $\eta $ and T are as sought.

Proof. (1) Fix a bijection $\pi :\kappa \leftrightarrow \kappa \times \kappa $ and let J be the corresponding ideal from Example 2.1. Fix a C-sequence $\vec C=\langle C_\delta \mid \delta <\kappa \rangle $ that avoids S, i.e., $\operatorname {\mathrm {acc}}(C_\delta )\cap S=\emptyset $ for every $\delta <\kappa $ . We shall walk along $\vec C$ , and let $\operatorname {\mathrm {tr}}$ , $\lambda $ , etc. be the induced characteristic maps. Define a colouring $c:[\kappa ]^2\rightarrow \kappa $ as follows. Given $\eta <\beta <\kappa $ , consider $(\eta _0,\eta _1):=\pi (\eta )$ , and then set $c(\eta ,\beta ):=\min (\operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _0,\beta ))\cap \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _1,\beta )))$ , provided that the latter is well-defined. Otherwise, just let $c(\eta ,\beta ):=0$ .

Let $I \in \mathcal I$ , and we shall show that c witnesses $\operatorname {{\mathsf{unbounded}}}^+(J^+,I,\text {NS}_\kappa \restriction (S\cap T^*(I)))$ . Let $A\in J^+$ and $B\in I^+$ be given. Consider $A_0:=\{\eta _0<\kappa \mid \sup \{\eta _1<\kappa \mid \pi ^{-1}(\eta _0,\eta _1)\in A\}=\kappa \}$ , and note that, since $A\in J^+$ , the following set is a club in $\kappa $ :

$$ \begin{align*}E:=\{\delta\in\operatorname{\mathrm{acc}}^+(A_0)\mid \forall \eta_0\in A_0\cap\delta\, \sup\{\eta_1<\delta\mid \pi^{-1}(\eta_0,\eta_1)\in A\cap\delta\}=\delta\}.\end{align*} $$

Claim 6.5.1. The following holds:

$$ \begin{align*}S\cap T^*(I)\cap E\subseteq \{ \delta<\kappa\mid \exists \eta\in A\cap\delta\, \{ \beta\in B\mid c(\eta,\beta)=\delta\}\in I^+\}.\end{align*} $$

Proof. Fix $\delta \in S\cap T^*(I)\cap E$ . As I is $\operatorname {\mathrm {cf}}(\delta )$ -indecomposable, we may fix some $\epsilon <\delta $ for which the following set is in $I^+$ :

$$ \begin{align*}B':=\{\beta\in B\setminus(\delta+1)\mid \lambda(\delta,\beta)\le\epsilon\}.\end{align*} $$

To justify, note that as $\vec C$ avoids S and as $\delta \in S$ , for every $\beta \in B \setminus (\delta +1)$ , $\lambda (\delta , \beta ) < \delta $ ; and secondly, as $B\setminus (\delta +1)$ is in $I^+$ and I is $\operatorname {\mathrm {cf}}(\delta )$ -indecomposable, such an $\epsilon < \delta $ can indeed be found.

Pick $\eta _0\in A_0\cap \delta $ above $\epsilon $ , and then pick an $\eta _1$ with $\min (C_\delta \setminus \eta _0)<\eta _1<\delta $ such that $\eta :=\pi ^{-1}(\eta _0,\eta _1)$ is in $A\cap \delta $ . For every $\beta \in B'$ ,

$$ \begin{align*}\lambda(\delta,\beta)\le\epsilon<\eta_0<\eta_1<\delta<\beta\end{align*} $$

and hence $\delta \in \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _0,\,\beta ))\cap \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _1,\,\beta ))$ . In addition,

$$ \begin{align*}\min(C_{\delta}\setminus\eta_0)<\eta_1\le\min(C_{\delta}\setminus\eta_1),\end{align*} $$

and hence $c(\eta ,\,\beta )=\delta $ .

The conclusion now follows from Fodor’s lemma.

(2) By Clause (1) and Proposition 6.2(2), it suffices to prove that $\operatorname {{\mathsf{onto}}}^{++}(J^{\text {bd}}[\kappa ], \vartheta )$ holds for every cardinal $\vartheta <\kappa $ . Now, since $\kappa $ admits a nonreflecting stationary set, the main result of [Reference RinotRin14] implies that $\operatorname {\mathrm {Pr}}_1(\kappa ,\kappa ,\kappa ,\omega )$ holds (see Definition 7.3). In particular, $\kappa \nrightarrow [\kappa ;\kappa ]^2_\vartheta $ holds for every $\vartheta \le \kappa $ . By [Reference Inamdar and RinotIR24, Theorem 5.4(2)], then, $\operatorname {{\mathsf{onto}}}^{++}(J^{\text {bd}}[\kappa ],\vartheta )$ holds for every $\vartheta <\kappa $ , as sought.

(3) By Clause (1), and Proposition 6.2(1), recalling [Reference Inamdar and RinotIR23, Corollary 8.6].

Proof. Fix a bijection $\pi :\kappa \leftrightarrow \kappa \times \kappa $ and let J be the corresponding ideal from Example 2.1. Let $\Theta \in [\operatorname {\mathrm {Reg}}(\kappa )]^{<\kappa }$ . By [Reference Meir Brodsky and RinotBR19, Lemma 2.5], we may fix a witness ${\vec {\mathcal C}=\langle \mathcal C_\delta \mid \delta <\kappa \rangle }$ for $\square (\kappa ,{<}\mu )$ such that for every club $D\subseteq \kappa $ , for every $\theta \in \Theta $ , there exists a $\delta \in E^\kappa _\theta $ such that $\sup (\operatorname {\mathrm {nacc}}(C)\cap D)=\delta $ for all $C\in \mathcal C_\delta $ . Let $\vec C=\langle C_\delta \mid \delta <\kappa \rangle $ be a transversal for $\vec {\mathcal C}$ . We shall walk along $\vec C$ , and let $\operatorname {\mathrm {tr}}$ , $\lambda _2$ , etc. be the induced characteristic maps. Define a colouring $c:[\kappa ]^2\rightarrow \kappa $ as follows. Given $\eta <\beta <\kappa $ , consider $(\eta _0,\eta _1):=\pi (\eta )$ . Let $\gamma :=\min (\operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _0,\beta ))\cap \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _1,\beta )))$ , and then set $c(\eta ,\beta ):=\min (C_\gamma \setminus \eta _1)$ , provided that the latter is well-defined. Otherwise, just let $c(\eta ,\beta ):=0$ .

To see that c witnesses $\operatorname {{\mathsf{unbounded}}}^+(J^+,\bigcup _{\theta \in \Theta }\mathcal I^\kappa _\theta \cap \mathcal J^\kappa _\mu ,\text {NS}_\kappa )$ , let $A\in J^+$ , $\theta \in \Theta $ , $I\in \mathcal I^\kappa _\theta \cap \mathcal I^\kappa _\mu $ and $B\in I^+$ be given. Derive from A the sets $A_0$ and E as in the proof of Lemma 6.5.

Claim 6.6.1. The following set is stationary:

$$ \begin{align*}T:=\{ \tau<\kappa\mid \exists \eta\in A\cap\tau\, \{ \beta\in B\mid c(\eta,\beta)=\tau\}\in I^+\}.\end{align*} $$

Proof. Let D be an arbitrary club in $\kappa $ , and we shall find a $\tau $ in $D\cap T$ . Fix $\delta \in E^\kappa _\theta $ such that $\sup (\operatorname {\mathrm {nacc}}(C)\cap D\cap E)=\delta $ for all $C\in \mathcal C_\delta $ . As I is $\mu $ -complete, we may fix a $C\in \mathcal C_\delta $ for which the following set is in $I^+$ :

$$ \begin{align*}B':=\{\beta\in B\setminus(\delta+1)\mid C_{\eth_{\delta,\beta}}\cap\delta=C\}.\end{align*} $$

As $\operatorname {\mathrm {cf}}(\delta )=\theta $ and as I is $\theta $ -indecomposable, we may fix some $\epsilon <\delta $ for which the following set is in $I^+$ :

$$ \begin{align*}B":=\{\beta\in B'\mid \lambda_2(\delta,\beta)\le\epsilon\}.\end{align*} $$

Pick $\eta _0\in A_0\cap \delta $ above $\epsilon $ . Fix $\tau \in \operatorname {\mathrm {nacc}}(C)\cap D\cap E$ above $\min (C\setminus \eta _0)$ . As $\tau \in E$ , we may then fix an $\eta _1$ with $\sup (C\cap \tau )<\eta _1<\tau $ such that $\eta :=\pi ^{-1}(\eta _0,\eta _1)$ is in $A\cap \tau $ . For every $\beta \in B"$ ,

$$ \begin{align*}\lambda(\eth_{\delta,\beta},\beta)\le\epsilon<\eta_0<\eta_1<\tau<\delta<\beta,\end{align*} $$

and hence $\eth _{\delta ,\beta }\in \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _0,\beta ))\cap \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _1,\beta ))$ . In addition,

$$ \begin{align*}\min(C_{\eth_{\delta,\beta}}\setminus\eta_0)\le\sup(C\cap\tau)<\eta_1\le\min(C_{\eth_{\delta,\beta}}\setminus\eta_1),\end{align*} $$

and hence $\min (\operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _0,\beta ))\cap \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _1,\beta )))=\eth _{\delta ,\beta }$ from which it follows that

$$ \begin{align*}c(\eta,\beta)=\min(C_{\eth_{\delta,\beta}}\setminus\eta_1)=\min(C\setminus\eta_1)=\tau,\end{align*} $$

as sought.

The conclusion now follows from Fodor’s lemma.

Proof. (1) Let $\theta ,\, \vartheta \in \operatorname {\mathrm {Reg}}(\kappa )$ . By the upcoming Clause (2), we may assume that $\kappa>\vartheta ^+$ . So, by Shelah’s club-guessing theorem [Reference ShelahShe94b, Claim 2.7] $\mathcal C(\kappa ,\vartheta )=\kappa $ . Now, the conclusion follows from Lemma 6.6 and Lemma 2.4(4).

(2) By Lemma 6.6, in particular, $\operatorname {{\mathsf{unbounded}}}^+(J^+, \mathcal I,J^{\text {bd}}[\kappa ])$ holds for some ideal $J\in \mathcal J^{\kappa }_\kappa $ . So, by Proposition 6.2(2), it suffices to prove that $\operatorname {{\mathsf{onto}}}^{++}(J^{\text {bd}}[\kappa ],\vartheta )$ holds. The latter follows from [Reference Inamdar and RinotIR24, Theorem 5.3(2)] and Lemma 7.2 below.

(3) By Proposition 6.2(1), it suffices to prove that $\operatorname {{\mathsf{onto}}}(J^{\text {bd}}[\kappa ],\kappa )$ holds. By [Reference Inamdar and RinotIR23, Lemma 8.3], for every successor cardinal $\kappa $ , implies $\operatorname {{\mathsf{onto}}}(J^{\text {bd}}[\kappa ],\kappa )$ .

(4) Suppose that $\diamondsuit (S)$ holds for some stationary $S\subseteq \kappa $ that does not reflect at regulars. By [Reference Inamdar and RinotIR23, Corollary 8.6], then, $\operatorname {{\mathsf{onto}}}(J^{\text {bd}}[\kappa ],\kappa )$ holds. Thus, letting $\theta \in \operatorname {\mathrm {Reg}}(\kappa )$ , we get from Lemma 6.6 and Proposition 6.2(1) that $\operatorname {{\mathsf{onto}}}^+(\mathcal I_\theta ^\kappa ,\kappa )$ holds.

Proof. Fix a bijection $\pi :\kappa \leftrightarrow \kappa \times \kappa $ and let J be the corresponding ideal from Example 2.1. Fix a C-sequence $\vec C=\langle C_\delta \mid \delta <\kappa \rangle $ , a cardinal $\nu <\mu $ , and a sequence $\langle S_i\mid i<\nu \rangle $ of stationary subsets of $T^*(I)$ such that for every $\delta \in \operatorname {\mathrm {acc}}(\kappa )$ , for some $i<\nu $ , $\operatorname {\mathrm {acc}}(C_\delta )\cap S_i=\emptyset $ . We shall walk along $\vec C$ , and let $\operatorname {\mathrm {tr}}$ , $\lambda _2$ , etc. be the induced characteristic maps. Define $c:[\kappa ]^2\rightarrow \kappa $ as follows. Given $\eta <\beta <\kappa $ , consider $(\eta _0,\eta _1):=\pi (\eta )$ . Let $\gamma :=\min (\operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _0,\beta ))\cap \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _1,\beta )))$ , and then set $c(\eta ,\beta ):=\min (C_\gamma \setminus \eta _1)$ , provided that the latter is well-defined. Otherwise, just let $c(\eta ,\beta ):=0$ .

To see that c witnesses , let $A\in J^+$ and $B\in I^+$ be given. Derive from A the sets $A_0$ and E as in the proof of Lemma 6.5.

Claim 6.10.1. The following set is stationary:

$$ \begin{align*}T:=\{\tau<\kappa\mid \exists\eta\in A\cap\tau\,\exists\delta<\kappa\, \{ \beta\in B\mid \tau\le c(\eta,\beta)<\delta\}\in I^+\}.\end{align*} $$

Proof. Let D be an arbitrary club in $\kappa $ , and we shall find a $\tau $ in $D\cap T$ . By possibly shrinking, we may assume that $D\subseteq E$ . Pick $\delta \in S_0\cap \bigcap _{i<\nu }\operatorname {\mathrm {acc}}^+(D\cap S_i)$ . For every $\beta \in B$ above $\delta $ , there is an $i<\nu $ such that $\operatorname {\mathrm {acc}}(C_{\eth _{\delta ,\beta }})\cap S_i=\emptyset $ . As $\nu <\mu $ and I is $\mu $ -complete, we may fix $i<\nu $ and $B'\subseteq B\setminus (\delta +1)$ in $I^+$ such that, for every $\beta \in B'$ , $\operatorname {\mathrm {acc}}(C_{\eth _{\delta ,\beta }})\cap S_i=\emptyset $ . As $\delta \in S_0$ , I is $\operatorname {\mathrm {cf}}(\delta )$ -indecomposable, so we may find a large enough $\eta _0\in A_0\cap \delta $ such that the following set is in $I^+$ :

$$ \begin{align*}B":=\{\beta\in B'\mid \lambda_2(\delta,\beta)<\eta_0\}.\end{align*} $$

As I is $\operatorname {\mathrm {cf}}(\delta )$ -indecomposable, we may next find some $\varepsilon <\delta $ such that the following set is in $I^+$ :

$$ \begin{align*}B"':=\{\beta\in B"\mid \min(C_{\eth_{\delta,\beta}}\setminus\eta_0)\le\varepsilon\}.\end{align*} $$

Pick $\tau \in S_i\cap D\cap \delta $ above $\varepsilon $ . For every $\beta \in B"'$ , as $\operatorname {\mathrm {acc}}(C_{\eth _{\delta ,\beta }})\cap S_i=\emptyset $ , it is the case that $\sup (C_{\eth _{\delta ,\beta }}\cap \tau )<\tau $ . As $\tau \in S_i\cap E$ , I is $\operatorname {\mathrm {cf}}(\tau )$ -indecomposable and we may find a large enough $\eta _1<\tau $ with $\pi ^{-1}(\eta _0,\eta _1)\in A\cap \tau $ such that the following set is in $I^+$ :

$$ \begin{align*}B"":=\{\beta\in B"'\mid \sup(C_{\eth_{\delta,\beta}}\cap\tau)<\eta_1\}.\end{align*} $$

Let $\eta <\tau $ be such that $\pi (\eta )=(\eta _0,\eta _1)$ . For every $\beta \in B""$ ,

$$ \begin{align*}\lambda(\eth_{\delta,\beta},\beta)<\eta_0<\eta_1<\tau<\delta\le\eth_{\delta,\beta}\le\beta\end{align*} $$

and hence $\eth _{\delta ,\beta }\in \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _0,\beta ))\cap \operatorname {\mathrm {Im}}(\operatorname {\mathrm {tr}}(\eta _1,\beta ))$ . In addition,

$$ \begin{align*}\min(C_{\eth_{\delta,\beta}}\setminus\eta_0)\le\varepsilon<\tau\le\min(C_{\eth_{\delta,\beta}}\setminus\eta_1)<\delta,\end{align*} $$

and hence $c(\eta ,\beta )\in [\tau ,\delta )$ , as sought.

Let T be given by the claim. Pick two functions $f:T\rightarrow A$ and $g:T\rightarrow \kappa $ such that for every $\tau \in T$ , $f(\tau )<\tau <g(\tau )$ and