3 research outputs found
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
Bounds on degrees of p-adic separating polynomials
We study a discrete optimization problem introduced by Babai, Frankl, Kutin, and Stefankovic (2001), which provides bounds on degrees of polynomials with p-adically controlled behavior. Such polynomials are of particular interest because they furnish bounds on the size of set systems satisfying Frankl-Wilson-type conditions modulo prime powers, with lower degree polynomials providing better bounds. We elucidate the asymptotic structure of solutions to the optimization problem, and we also provide an improved method for finding solutions in certain circumstances