16,956 research outputs found
Almost-Fisher families
A classic theorem in combinatorial design theory is Fisher's inequality,
which states that a family of subsets of with all pairwise
intersections of size can have at most non-empty sets. One may
weaken the condition by requiring that for every set in , all but
at most of its pairwise intersections have size . We call such
families -almost -Fisher. Vu was the first to study the maximum
size of such families, proving that for the largest family has
sets, and characterising when equality is attained. We substantially refine his
result, showing how the size of the maximum family depends on . In
particular we prove that for small one essentially recovers Fisher's
bound. We also solve the next open case of and obtain the first
non-trivial upper bound for general .Comment: 27 pages (incluiding one appendix
Short proofs of three results about intersecting systems
In this note, we give short proofs of three theorems about intersection
problems. The first one is a determination of the maximum size of a nontrivial
-uniform, -wise intersecting family for , which improves upon a recent result of
O'Neill and Verstra\"{e}te. Our proof also extends to -wise,
-intersecting families, and from this result we obtain a version of the
Erd\H{o}s-Ko-Rado theorem for -wise, -intersecting families.
The second result partially proves a conjecture of Frankl and Tokushige about
-uniform families with restricted pairwise intersection sizes.
The third result concerns graph intersections. Answering a question of Ellis,
we construct -intersecting families of graphs which have size larger
than the Erd\H{o}s-Ko-Rado-type construction whenever is sufficiently large
in terms of .Comment: 12 pages; we added a new result, Theorem 1
Coloring non-crossing strings
For a family of geometric objects in the plane
, define as the least
integer such that the elements of can be colored with
colors, in such a way that any two intersecting objects have distinct
colors. When is a set of pseudo-disks that may only intersect on
their boundaries, and such that any point of the plane is contained in at most
pseudo-disks, it can be proven that
since the problem is equivalent to cyclic coloring of plane graphs. In this
paper, we study the same problem when pseudo-disks are replaced by a family
of pseudo-segments (a.k.a. strings) that do not cross. In other
words, any two strings of are only allowed to "touch" each other.
Such a family is said to be -touching if no point of the plane is contained
in more than elements of . We give bounds on
as a function of , and in particular we show that
-touching segments can be colored with colors. This partially answers
a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems
of strings.Comment: 19 pages. A preliminary version of this work appeared in the
proceedings of EuroComb'09 under the title "Coloring a set of touching
strings
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
Newton polygons and curve gonalities
We give a combinatorial upper bound for the gonality of a curve that is
defined by a bivariate Laurent polynomial with given Newton polygon. We
conjecture that this bound is generically attained, and provide proofs in a
considerable number of special cases. One proof technique uses recent work of
M. Baker on linear systems on graphs, by means of which we reduce our
conjecture to a purely combinatorial statement.Comment: 29 pages, 18 figures; erratum at the end of the articl
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