98 research outputs found
Tower-type bounds for unavoidable patterns in words
A word is said to contain the pattern if there is a way to substitute
a nonempty word for each letter in so that the resulting word is a subword
of . Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised
the patterns which are unavoidable, in the sense that any sufficiently long
word over a fixed alphabet contains . Zimin's characterisation says that a
pattern is unavoidable if and only if it is contained in a Zimin word, where
the Zimin words are defined by and . We
study the quantitative aspects of this theorem, obtaining essentially tight
tower-type bounds for the function , the least integer such that any
word of length over an alphabet of size contains . When , the first non-trivial case, we determine up to a constant factor,
showing that .Comment: 17 page
Unavoidable patterns in locally balanced colourings
Which patterns must a two-colouring of contain if each vertex has at
least red and blue neighbours? In this paper,
we investigate this question and its multicolour variant. For instance, we show
that any such graph contains a -blow-up of an \textit{alternating 4-cycle}
with .Comment: Improved expositio
Unavoidable patterns in complete simple topological graphs
In this paper, we show that every complete -vertex simple topological
graph contains a topological subgraph on at least
vertices that is weakly isomorphic to the complete convex geometric graph or
the complete twisted graph. This improves the previously known bound of
due to Pach, Solymosi, and T\'oth. We also show that
every complete -vertex simple topological graph contains a planar path of
length at least
Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations
It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP. It has also been conjectured that the length of a minimal solution to a quadratic equation is at most exponential in the length of the equation, with the latter conjecture implying the former. We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations. We also discuss a connection of our techniques to the topic of unavoidable patterns, and the possibility of exploiting this connection to produce further similar results
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