2 research outputs found
Monochromatic Hilbert cubes and arithmetic progressions
The Van der Waerden number denotes the smallest such that
whenever is --colored there exists a monochromatic arithmetic
progression of length . Similarly, the Hilbert cube number denotes
the smallest such that whenever is --colored there exists a
monochromatic affine --cube, that is, a set of the form for some and .
We show the following relation between the Hilbert cube number and the Van
der Waerden number. Let be an integer. Then for every ,
there is a such that Thus we improve upon state of the art lower bounds
for conditional on being significantly larger than . In
the other direction, this shows that the if the Hilbert cube number is close
its state of the art lower bounds, then is at most doubly exponential
in .
We also show the optimal result that for any Sidon set , one has \left|\left\{\sum_{b \in B} b : B \subseteq
A\right\}\right| = \Omega( |A|^3) .$
The largest projective cube-free subsets of
In the Boolean lattice, Sperner's, Erd\H{o}s's, Kleitman's and Samotij's
theorems state that families that do not contain many chains must have a very
specific layered structure. We show that if instead of we work
in , several analogous statements hold if one replaces the
word -chain by projective cube of dimension .
We say that is a projective cube of dimension if there are numbers
such that
As an analog of Sperner's and Erd\H{o}s's theorems, we show that whenever
is a power of two, the largest -cube free set in
is the union of the largest layers. As an analog of
Kleitman's theorem, Samotij and Sudakov asked whether among subsets of
of given size , the sets that minimize the number of
Schur triples (2-cubes) are those that are obtained by filling up the largest
layers consecutively. We prove the first non-trivial case where ,
and conjecture that the analog of Samotij's theorem also holds.
Several open questions and conjectures are also given.Comment: 23 page