99 research outputs found
Semigroups with the Erdös-Turán Property
A set X in a semigroup G has the Erdös-Turán property ET if,
for any basis A of X, the representation function rA is ubounded,
where rA(x) counts the number of representations of x as a product
two elements in A. We show that, under some conditions, operations
on binary vectors whose value at each coordinate depends only on
neighbouring coordinates of the factors give rise to semigroups with
the ET{property. In particular countable powers of semigroups with
no mutually inverse elements have the ET{property. As a consequence,
for each k there is N(k) such that, for every ¯nite subset X of a group
G with X \ X¡1 = f1g, the representation function of every basis of
XN ½ GN, N ¸ N(k), is not bounded by k. This is in contrast with
the known fact that each p{elementary group admits a basis of the
whole group whose representation function is bounded by an absolute
constan
Rainbow-free 3-colorings in abelian groups
A 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic
progression with its members having pairwise distinct colors. We describe
the structure of rainbow–free colorings of abelian groups. This structural description
proves a conjecture of Jungi´c et al. on the size of the smallest chromatic class of a rainbow–free coloring of cyclic groups.Postprint (published version
Extremal families for Kruskal-Katona Theorem
Given a set of size and a positive integer , Kruskal--Katona theorem gives the minimum size of the shadow of a family of -sets of in terms of the cardinality of . We give a characterization of the families of -sets satisfying equality in Kruskal--Katona theorem. This answers a question of F\"uredi and Griggs.Peer ReviewedPostprint (published version
On a problem by Shapozenko on Johnson Graphs
The final publication is available at Springer via http://dx.doi.org/10.1007/s00373-018-1923-7The Johnson graph J(n, m) has the m-subsets of {1,2,…,n} as vertices and two subsets are adjacent in the graph if they share m-1 elements. Shapozenko asked about the isoperimetric function µn,m(k) of Johnson graphs, that is, the cardinality of the smallest boundary of sets with k vertices in J(n, m) for each 1=k=(nm) . We give an upper bound for µn,m(k) and show that, for each given k such that the solution to the Shadow Minimization Problem in the Boolean lattice is unique, and each sufficiently large n, the given upper bound is tight. We also show that the bound is tight for the small values of k=m+1 and for all values of k when m=2 .Peer ReviewedPostprint (author's final draft
Punctured combinatorial Nullstellensätze
In this article we present a punctured version of Alon's Nullstellensatz which states that if vanishes at nearly all, but not all, of the common zeros of some polynomials then every -residue of , where the ideal , has a large degree.
Furthermore, we extend Alon's Nullstellensatz to functions which have multiple zeros at the common zeros of and prove a punctured version of this generalised version.
Some applications of these punctured Nullstellens\"atze to projective and affine geometries over an arbitrary field are considered which, in the case that the field is finite, will lead to some bounds related to linear codes containing the all one vector
Set systems with distinct sumsets
A family of -subsets of is a Sidon system if the sumsets , are pairwise distinct.
We show that the largest cardinality of a Sidon system of -subsets of satisfies and the asymptotic lower bound .
More precise bounds on are obtained for .
We also obtain the threshold probability for a random system to be Sidon for and .Peer ReviewedPostprint (author's final draft
On some groups related to arc-colored digraphs
An arc-coloring of a digraph G is an assignation of colors to its arcs in such
a way that all arcs incident to each vertex, as well as all arcs incident from
each vertex, have different colors. In an arc-colored d-regular digraph, each
color can be associated to a permutation of the vertices of the digraph. In
this paper we show the relation between the permutation group generated
by these permutations and the groups of those automorphisms of the arccolored
digraph which preserve or exchange colors.Postprint (published version
Rainbow perfect matchings in r-partite graph structures
A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer ReviewedPostprint (author's final draft
Counting configuration-free sets in groups
© 2017 Elsevier Ltd. We provide asymptotic counting for the number of subsets of given size which are free of certain configurations in finite groups. Applications include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. Random sparse versions and threshold probabilities for existence of configurations in sets of given density are presented as well.Postprint (updated version
General properties of c-circulant digraphs
A digraph is said to be a c-circulant if its adjacency matrix is c-circulant. This paper deals with general properties of this family of digraphs, as isomorphisms, regularity, strong connectivity, diameter and the relation between c-circulant digraphs and the line digraph technique.Peer ReviewedPostprint (published version
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