TITLE: Über theorem on unavoidable patterns?
QUESTION [8 upvotes]: Let $A$ be an alphabet of $k$ symbols,
and $p$ a pattern.
An example of a pattern is $p=XX$, where $X$ is any finite
string of symbols from $A^+$.
Avoiding $p$ is avoiding any subword repeated twice in a row.
Such strings are called square-free.
It is well known that there are no infinite binary ($k=2$) strings
of symbols that are square-free
(in fact, only $0$, $1$, $01$, $10$, $010$, and $101$ are square-free),
but there are infinite ternary ($k=3$) square-free strings,
as proved by Axel Thue.
Other examples:
the patterns $X$ and $XYX$ are unavoidable on any alphabet.
My question is:

Q. Is there a theorem of the form: Any alphabet $A$ with $|A| \le k$
  cannot avoid any patterns $p$ of the form [some description of these patterns $p$
  as a function of $k$],
  i.e., there are no infinite strings that avoid these $p$?

In other words, is there a pattern to—a characterization of—the patterns that are unavoidable,
for a given $k$? Or are there, to date, only claims that
specific patterns are unavoidable?

REPLY [7 votes]: According to the 2013 paper "Computing the Partial Word Avoidability Indices of
Ternary Patterns" by Blanchet-Sadri, Lohr, and Scott,

The problem of deciding whether a given pattern is avoidable has been
  solved [1, 14], but the one of deciding whether it is k-avoidable has remained
  open.

So, it seems to be an open problem.