## Abstract

The Dushnik–Miller dimension of a poset *P* is the least *d* for which *P* can be embedded into a product of *d* chains. Lewis and Souza isibility order on the interval of integers \([N/\kappa , N]\) is bounded above by \(\kappa (\log \kappa )^{1+o(1)}\) and below by \(\Omega ((\log \kappa /\log \log \kappa )^2)\). We improve the upper bound to \(O((\log \kappa )^3/(\log \log \kappa )^2).\) We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.

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## Acknowledgements

I would like to thank Noah Kravitz for helpful conversations and detailed feedback on earlier versions of this paper. Additionally, I am grateful to Victor Souza for bringing other divisibility orders to my attention, among other helpful comments. I would also like to thank Michael Ren and William Trotter for helpful conversations and Joe Gallian, Carl Schildkraut, Zach Hunter, and the anonymous reviewers for comments on earlier versions of this paper. Finally, I want to express my gratitude to Joe Gallian, Amanda Burcroff, Colin Defant, Noah Kravitz, and Yelena Mandelshtam for organizing the Duluth REU and the project suggestion.

## Funding

Open Access funding provided by the MIT Libraries. This research was conducted at the University of Minnesota Duluth Mathematics REU and was supported, in part, by NSF-DMS Grant 1949884 and NSA Grant H98230-20-1-0009. Additional support was provided by the CYAN Mathematics Undergraduate Activities Fund.

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This is a single-author paper. All contributions are due to Milan Haiman.

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## Appendices

### Appendix A. Computations for Theorem 1.7

In this appendix we show the following bound.

### Proposition A.1

Let \(\textbf{v}=(\log 2,\log 3, \dots , \log p_n)\), where \(p_i\) is the *i*th prime. Let \(\kappa \ge 3\) be a real number, and let \(r=4\frac{\log \kappa }{\log \log \kappa }\). Then \(m(\textbf{v},r)\ge 2\log (\kappa )\).

### Proof

We use [10], which provides bounds on the Chebyshev function \(\vartheta (x)=\sum _{p\le x}\log p\) (where the sum is over primes *p*). Note that \(m(\textbf{v},r)=\vartheta (p_{\lfloor {r}\rfloor })\). By [10, Theorem 6], we have that

Now, let \(\lambda =\log \kappa >1\) and note that \(\frac{\lambda }{\log \lambda }\ge e\). So

Let \(\eta =1.076869\) and note that \(\eta <\log (3.5)\). Using the above bounds and \(\frac{\log \log \lambda }{\log \lambda }\le \frac{1}{e}\), we have that

So \(m(\textbf{v},r)\ge 2\log \kappa \), as desired.\(\square \)

### Appendix B. Computations for Theorem 1.9

In this appendix we show bounds on irreducible polynomials in \(\mathbb {F}_q[x]\). Let \(n_i\) be the number of monic irreducible polynomials of degree *i*. Recall that the product of all monic irreducible polynomials of degree dividing *i* is \(x^{q^i}-x\). In particular, this means that \(n_i\le q^i/i\). In the notation of Section 6, we have that \(n=n_1+\dots +n_{\delta }\) and \(\textbf{v}\in \mathbb {R}^n\) is a vector with \(n_i\) entries being *i*.

### Proposition B.1

We have \(n\le q^\delta \).

### Proof

We have \(n_1=q\) and for \(i\ge 2\),

So

\(\square \)

### Proposition B.2

Let \(r=4.6\frac{\delta \log q}{\log \delta }\). Then \(m(\textbf{v},r)\ge 2\delta \).

### Proof

For each *i*, \(\textbf{v}\) has \(n_1+\dots +n_{i-1}\) entries that are less than *i*. Thus \(m(\textbf{v},r)\) has

terms that are at least *i*. Then for any \(d\in \mathbb {N}\) we have

We will apply this with \(d=\lfloor {\log \delta /\log q}\rfloor +1\). Since \(\frac{\delta }{\log \delta }\ge e\) and \(\log q\ge \log 2\),

Now we have

By rearrangement,

So we have

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Haiman, M. The Dimension of Divisibility Orders and Multiset Posets.
*Order* (2023). https://doi.org/10.1007/s11083-023-09653-7

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DOI: https://doi.org/10.1007/s11083-023-09653-7