% Here is the LateX version of the letter I wrote to you.
% Please feel free to use any part of it in the updated version of
% your book.
% A.Miller
\documentclass[12pt]{article}
\usepackage{amssymb}
\def\infsets{[\omega]^\omega} \def\res{|_}
\def\qed{\nopagebreak\par\noindent\nopagebreak $\Box$ \par}
\def\proof{{\par\noindent{\bf Proof: }}}
\def\forces{\mid\vdash}
\newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma}
\begin{document}
\begin{flushright}
Arnold W. Miller\\
Department of Mathematics\\
University of Wisconsin\\
Madison, WI 53706\\
July 1, 1987 \\
\end{flushright}
\bigskip
\hrule
\bigskip
\noindent Dear Professor Fra\"{i}ss\'{e}
I enjoyed very much reading your book, {\bf Theory of Relations}.
Thank you for writting it.
Probably someone has already answered the problem of
Hagendorf\footnote{I have sent copies of this letter to Hagendorf, Kunen,
Pouzet, and Veli\v{c}kovi\'{c}.}
which you mention on page 136:
\begin{quote}
Existence of a strictly decreasing $\omega_1$-sequence of denumerable
partial orderings.
\end{quote}
The following result, which I proved jointly with Ken Kunen,
answers this question in the affirmative.
\begin{theorem}
There exists $\langle P_X: X\in\infsets\rangle$
where each $P_X$ is a countable
poset and
$$P_X \mbox{ embeds into } P_Y \iff X\subseteq^* Y$$
where $\infsets$ is the set of infinite subsets of $\omega=\{0,1,2,\ldots\}$
and $X\subseteq^* Y$ means inclusion mod finite, i.e. $X\setminus Y$ is finite.
\end{theorem}
Since there are decreasing mod finite $\omega_1$ sequences in $\infsets$
we get that the same is true for countable posets under embedding.
\begin{lemma}
There is a set $\langle C_n: n\in\omega\rangle$ of finite partial orders
such for any $n\in\omega$, $C_n$ cannot be embedded in the disjoint
union of $\{C_m: m\neq n\}$. Furthermore no $C_n$ contains a chain
of length three.
\end{lemma}
\proof
For $n\geq 3$ let $C_{n-3}$ be the following ordering on $2n$ points
$$\{a_i,b_i: i