3,483 research outputs found
The number of subsets of integers with no -term arithmetic progression
Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely
many values of , the number of subsets of that do not
contain a -term arithmetic progression is at most , where
is the maximum cardinality of a subset of without
a -term arithmetic progression. This bound is optimal up to a constant
factor in the exponent. For all values of , we prove a weaker bound, which
is nevertheless sufficient to transfer the current best upper bound on
to the sparse random setting. To achieve these bounds, we establish a new
supersaturation result, which roughly states that sets of size
contain superlinearly many -term arithmetic progressions.
For integers and , Erd\Ho s asked whether there is a set of integers
with no -term arithmetic progression, but such that any -coloring
of yields a monochromatic -term arithmetic progression. Ne\v{s}et\v{r}il
and R\"odl, and independently Spencer, answered this question affirmatively. We
show the following density version: for every and , there
exists a reasonably dense subset of primes with no -term arithmetic
progression, yet every of size contains a
-term arithmetic progression.
Our proof uses the hypergraph container method, which has proven to be a very
powerful tool in extremal combinatorics. The idea behind the container method
is to have a small certificate set to describe a large independent set. We give
two further applications in the appendix using this idea.Comment: To appear in International Mathematics Research Notices. This is a
longer version than the journal version, containing two additional minor
applications of the container metho
Maximising the number of cycles in graphs with forbidden subgraphs
Fix and let be a graph with containing a critical edge. We show that for sufficiently large the unique -vertex -free graph containing the maximum number of cycles is . This resolves both a question and a conjecture of Arman, Gunderson and Tsaturian \cite{Gund1}
On Approximating the Number of -cliques in Sublinear Time
We study the problem of approximating the number of -cliques in a graph
when given query access to the graph.
We consider the standard query model for general graphs via (1) degree
queries, (2) neighbor queries and (3) pair queries. Let denote the number
of vertices in the graph, the number of edges, and the number of
-cliques. We design an algorithm that outputs a
-approximation (with high probability) for , whose
expected query complexity and running time are
O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log
n,1/\varepsilon,k).
Hence, the complexity of the algorithm is sublinear in the size of the graph
for . Furthermore, we prove a lower bound showing that
the query complexity of our algorithm is essentially optimal (up to the
dependence on , and ).
The previous results in this vein are by Feige (SICOMP 06) and by Goldreich
and Ron (RSA 08) for edge counting () and by Eden et al. (FOCS 2015) for
triangle counting (). Our result matches the complexities of these
results.
The previous result by Eden et al. hinges on a certain amortization technique
that works only for triangle counting, and does not generalize for larger
cliques. We obtain a general algorithm that works for any by
designing a procedure that samples each -clique incident to a given set
of vertices with approximately equal probability. The primary difficulty is in
finding cliques incident to purely high-degree vertices, since random sampling
within neighbors has a low success probability. This is achieved by an
algorithm that samples uniform random high degree vertices and a careful
tradeoff between estimating cliques incident purely to high-degree vertices and
those that include a low-degree vertex
On two problems in Ramsey-Tur\'an theory
Alon, Balogh, Keevash and Sudakov proved that the -partite Tur\'an
graph maximizes the number of distinct -edge-colorings with no monochromatic
for all fixed and , among all -vertex graphs. In this
paper, we determine this function asymptotically for among -vertex
graphs with sub-linear independence number. Somewhat surprisingly, unlike
Alon-Balogh-Keevash-Sudakov's result, the extremal construction from
Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of
distinct edge-colorings with no monochromatic cliques among all graphs with
sub-linear independence number, even in the 2-colored case.
In the second problem, we determine the maximum number of triangles
asymptotically in an -vertex -free graph with . The
extremal graphs have similar structure to the extremal graphs for the classical
Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page
A note on fractional covers of a graph
A fractional colouring of a graph is a function that assigns a
non-negative real value to all possible colour-classes of containing any
vertex of , such that the sum of these values is at least one for each
vertex. The fractional chromatic number is the minimum sum of the values
assigned by a fractional colouring over all possible such colourings of .
Introduced by Bosica and Tardif, fractional covers are an extension of
fractional colourings whereby the real-valued function acts on all possible
subgraphs of belonging to a given class of graphs. The fractional chromatic
number turns out to be a special instance of the fractional cover number. In
this work we investigate fractional covers acting on -clique-free
subgraphs of which, although sharing some similarities with fractional
covers acting on -colourable subgraphs of , they exhibit some
peculiarities. We first show that if a simple graph is a homomorphic
image of a simple graph , then the fractional cover number defined on the
-clique-free subgraphs of is bounded above by the corresponding
number of . We make use of this result to obtain bounds for the associated
fractional cover number of graphs that are either -colourable or
-colourable.Comment: 8 page
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