In his book [1], Paul Larson remarks (Remark 1.1.22) that in *L* there is a function $h:\omega\_1\rightarrow\omega\_1$ such that for any countable elementary submodel $X$ of $V\_\gamma$ (where $\gamma$ is the first strong limit cardinal), we have the order-type of $X\cap Ord$ is strictly less than $h(X\cap\omega\_1)$. 


What is known about the relationship between the existence of such a function and large cardinal phenomena? 


Larson remarks that the existence of such a function is consistent with many large cardinals, and later in the book sketches a result of Velickovic showing that no such function exists in the presence of a precipitous ideal on $\omega\_1$. What more is known? Happy to also learn results concerning other related functions, or be pointed to associated references.


[1] *Larson, Paul B.*, The stationary tower. Notes on a course by W. Hugh Woodin, University Lecture Series 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3604-8/pbk). x, 132 p. (2004). [ZBL1072.03031](https://zbmath.org/?q=an:1072.03031).



Consequences of existence of a certain function from $\omega_1$ to $\omega_1$
In the paper "[The consistency strength of the perfect set property for universally Baire sets of reals](https://arxiv.org/abs/1807.02213)", Ralf Schindler and I show that the negation of a very similar statement is equiconsistent with the existence of what we call a virtually Shelah cardinal.


We say that a cardinal $\kappa$ is *virtually Shelah* if for every function $f:\kappa \to \kappa$ there is an ordinal $\lambda > \kappa$, a transitive set $M$ with $V\_\lambda \subset M$, and a generic elementary embedding $j : V\_\lambda \to M$ with critical point $\kappa$ and $j(f)(\kappa) \le \lambda$.


(By "generic elementary embedding" we mean an elementary embedding between structures in $V$ that exists in some generic extension of $V$. By absoluteness it suffices to consider any generic extension in which the domain structure is countable.)


This is a fairly weak large cardinal hypothesis. If $0^\sharp$ exists, then every Silver indiscernible is a virtually Shelah cardinal in $L$. On the other hand, every virtually Shelah cardinal is ineffable and a limit of ineffable cardinals.


By Theorem 1.2 of our paper, the following statements are equiconsistent modulo ZFC:


1. There is a virtually Shelah cardinal.
2. Every universally Baire set of reals has the perfect set property.
3. For every function $f : \omega\_1 \to \omega\_1$ there is an ordinal $\lambda > \omega\_1$ such that for a stationary set of $\sigma \in \mathcal{P}\_{\omega\_1}(\lambda)$ we have $\sigma \cap \omega\_1 \in \omega\_1$ and the order type of $\sigma$ is at least $f(\sigma \cap \omega\_1)$.


More specifically, we showed:


* Statement 1 implies that after forcing with $\text{Col}(\omega,\mathord{<}\kappa)$ where $\kappa$ is a virtually Shelah cardinal, statement 2 holds.
* Statement 2 implies statement 3.
* Statement 3 implies that $\omega\_1^V$ is a virtually Shelah cardinal in $L$.


I would also be interested to hear what else is known about statement 3 or the related statement in your question. In the paper, we only considered statement 3 because it came up naturally as a convenient intermediary between statements 1 and 2.


