103 research outputs found
The parity search problem
We prove that for any positive integers and there exists a collection
consisting of subsets of such
that for any two distinct subsets and of whose size is at most
there is an index for which and
have different parity. Here we think of as fixed whereas is thought of
as tending to infinity, and the base of the logarithm is .
Translated into the language of combinatorial search theory, this tells us
that queries suffice to identify up to marked items from a
totality of items if the answers one gets are just whether an even or an
odd number of marked elements has been queried, even if the search is performed
non-adaptively. Since the entropy method easily yields a matching lower bound
for the adaptive version of this problem, our result is asymptotically best
possible.
This answers a question posed by D\'aniel Gerbner and Bal\'azs Patk\'os in
Gyula O.H. Katona's Search Theory Seminar at the R\'enyi institute
The Clique Density Theorem
Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts
that for any integer every graph on vertices with more than
edges contains a clique of size , i.e.,
mutually adjacent vertices. The corresponding extremal graphs are balanced
-partite graphs.
The question as to how many such -cliques appear at least in any
-vertex graph with edges has been intensively studied in the
literature. In particular, Lov\'{a}sz and Simonovits conjectured in the 1970s
that asymptotically the best possible lower bound is given by the complete
multipartite graph with edges in which all but one vertex class is
of the same size while the remaining one may be smaller.
Their conjecture was recently resolved for by Razborov and for by
Nikiforov. In this article, we prove the conjecture for all values of .Comment: 25 pages, second version addresses changes arising from the referee
report
Counting odd cycles in locally dense graphs
We prove that for any given and , every
sufficiently large -dense graph contains for each odd
integer at least cycles of length . Here,
being -dense means that every set containing at
least~ vertices spans at least edges,
and what we really count is the number of homomorphisms from an -cycle into
.
The result adresses a question of Y. Kohayakawa, B. Nagle, V. R\"odl, and M.
Schacht
Maximum star densities
Given an integer and a real number , which
graphs of edge density contain the largest number of -edge stars?
For Ahlswede and Katona proved that asymptotically there cannot be more
such stars than in a clique or in the complement of a clique (depending on the
value of ). Here we extend their result to all integers .Comment: third version addresses changes arising from the referee report
Decomposing graphs into forests
We give a simple graph-theoretic proof of a classical result due to C. St. J.
A. Nash-Williams on covering graphs by forests. Moreover we derive a slight
generalisation of this statement where some edges are preassigned to distinct
forests.Comment: This version differs slightly from the version published by the
Journa
Forcing quasirandomness with triangles
We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson
initiated the study of families of graphs with the property that
if a large graph has approximately homomorphism density for some
fixed for every , then is quasirandom with
density . Such families are said to be forcing. Several forcing
families were found over the last three decades and characterising all
bipartite graphs such that is a forcing pair is a well-known open
problem in the area of quasirandom graphs, which is closely related to
Sidorenko's conjecture. In fact, most of the known forcing families involve
bipartite graphs only.
We consider forcing pairs containing the triangle . In particular, we
show that if is a forcing pair, then so is , where is
obtained from by replacing every edge of by a triangle (each of which
introduces a new vertex). For the proof we first show that is a
forcing pair, which strengthens related results of Simonovits and S\'os and of
Conlon et al.Comment: 16 pages, second version addresses changes arising from the referee
report
Weighted variants of the Andr\'asfai-Erd\H{o}s-S\'os Theorem
A well known result due to Andr\'asfai, Erd\H{o}s, and S\'os asserts that for
every -free graph on vertices with
is -partite. We study related questions in
the context of weighted graphs, which are motivated by recent work on the
Ramsey-Tur\'an problem for cliques.Comment: 8 figures; revised according to referee repor
A tale of stars and cliques
We show that for an infinitely many natural numbers there are -uniform
hypergraphs which admit a `rescaling phenomenon' as described in [9]. More
precisely, let denote the class of -graphs on
vertices in which the sizes of all pairwise intersections of edges belong to a
set . We show that if for some and , and~ is
chosen in some special way, the densest graphs in are
either dominated by stars of large degree, or basically, they are `-thick'
-graphs in which vertices are partitioned into groups of vertices
each and every edge is a union of such groups. It is easy to see that,
unlike in stars, the maximum degree of -thick graphs is of a lower order
than the number of its edges. Thus, if we study the graphs from
with a prescribed number of edges which minimize
the maximum degree, around the value of which is the number of edges of the
largest -thick graph, a rapid, discontinuous phase transition can be
observed. Interestingly, these two types of -graphs determine the structure
of all hypergraphs in . Namely, we show that each such
hypergraph can be decomposed into a -thick graph , a special collection
of stars, and a sparse `left-over' graph .Comment: second version addresses changes arising from the referee report
The chromatic number of finite type-graphs
By a finite type-graph we mean a graph whose set of vertices is the set of
all -subsets of for some integers , and
in which two such sets are adjacent if and only if they realise a certain order
type specified in advance. Examples of such graphs have been investigated in a
great variety of contexts in the literature with particular attention being
paid to their chromatic number. In recent joint work with Tomasz {\L}uczak, two
of the authors embarked on a systematic study of the chromatic numbers of such
type-graphs, formulated a general conjecture determining this number up to a
multiplicative factor, and proved various results of this kind. In this article
we fully prove this conjecture.Comment: second version addresses changes arising from the referee report
Hypergraphs with vanishing Tur\'an density in uniformly dense hypergraphs
P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel
Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs
that have to appear in any sufficiently large hypergraph of positive
density. We study related questions for -uniform hypergraphs with the
additional assumption that has to be uniformly dense with respect to vertex
sets. In particular, we characterise those hypergraphs that are guaranteed
to appear in large uniformly dense hypergraphs of positive density. We also
review the case when the density of the induced subhypergraphs of may
depend on the proportion of the considered vertex sets.Comment: 26 pages, this version addresses changes arising from the referee
report
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