376 research outputs found
On Universal Cycles for new Classes of Combinatorial Structures
A universal cycle (u-cycle) is a compact listing of a collection of
combinatorial objects. In this paper, we use natural encodings of these objects
to show the existence of u-cycles for collections of subsets, matroids,
restricted multisets, chains of subsets, multichains, and lattice paths. For
subsets, we show that a u-cycle exists for the -subsets of an -set if we
let vary in a non zero length interval. We use this result to construct a
"covering" of length for all subsets of of size
exactly with a specific formula for the term. We also show that
u-cycles exist for all -length words over some alphabet which
contain all characters from Using this result we provide
u-cycles for encodings of Sperner families of size 2 and proper chains of
subsets
Sperner systems with restricted differences
Let be a family of subsets of and be a subset of
. We say is an -differencing Sperner system if
for any distinct . Let be a prime
and be a power of . Frankl first studied -modular -differencing
Sperner systems and showed an upper bound of the form
. In this paper, we obtain new upper bounds on
-modular -differencing Sperner systems using elementary -adic analysis
and polynomial method, extending and improving existing results substantially.
Moreover, our techniques can be used to derive new upper bounds on subsets of
the hypercube with restricted Hamming distances. One highlight of the paper is
the first analogue of the celebrated Snevily's theorem in the -modular
setting, which results in several new upper bounds on -modular -avoiding
-intersecting systems. In particular, we improve a result of Felszeghy,
Heged\H{u}s, and R\'{o}nyai, and give a partial answer to a question posed by
Babai, Frankl, Kutin, and \v{S}tefankovi\v{c}.Comment: 22 pages, results in table 1 and section 6.1 improve
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