621 research outputs found
The anti-Ramsey threshold of complete graphs
For graphs and , let G {\displaystyle\smash{\begin{subarray}{c}
\hbox{\tiny\rm rb} \\ \longrightarrow \\ \hbox{\tiny\rm p}
\end{subarray}}}H denote the property that for every proper edge-colouring of
there is a rainbow in . It is known that, for every graph , an
asymptotic upper bound for the threshold function of this property for the random graph is
, where denotes the so-called maximum
-density of . Extending a result of Nenadov, Person, \v{S}kori\'c, and
Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower
bound for for . Furthermore, we show that .Comment: 19 page
Graph removal lemmas
The graph removal lemma states that any graph on n vertices with o(n^{v(H)})
copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite
its innocent appearance, this lemma and its extensions have several important
consequences in number theory, discrete geometry, graph theory and computer
science. In this survey we discuss these lemmas, focusing in particular on
recent improvements to their quantitative aspects.Comment: 35 page
Thresholds for constrained Ramsey and anti-Ramsey problems
Let and be graphs. A graph has the constrained Ramsey
property for if every edge-colouring of contains either a
monochromatic copy of or a rainbow copy of . Our main result gives a
0-statement for the constrained Ramsey property in whenever for some and is not a forest. Along with previous work
of Kohayakawa, Konstadinidis and Mota, this resolves the constrained Ramsey
property for all non-trivial cases with the exception of , which
is equivalent to the anti-Ramsey property for .
For a fixed graph , we say that has the anti-Ramsey property for
if any proper edge-colouring of contains a rainbow copy of . We show
that the 0-statement for the anti-Ramsey problem in can be reduced to
a (necessary) colouring statement, and use this to find the threshold for the
anti-Ramsey property for some particular families of graphs.Comment: 27 page
Large rainbow cliques in randomly perturbed dense graphs
For two graphs and , write if has the property that every {\sl proper} colouring of its edges
yields a {\sl rainbow} copy of .
We study the thresholds for such so-called {\sl anti-Ramsey} properties in
randomly perturbed dense graphs, which are unions of the form , where is an -vertex graph with edge-density at least
, and is a constant that does not depend on .
Our results in this paper, combined with our results in a companion paper,
determine the threshold for the property for every . In this paper, we
show that for the threshold is ; in fact, our -statement is a supersaturation result. This
turns out to (almost) be the threshold for as well, but for every , the threshold is lower; see our companion paper for more details.
In this paper, we also consider the property , and show that the
threshold for this property is for every ; in particular,
it does not depend on the length of the cycle . It is worth
mentioning that for even cycles, or more generally for any fixed bipartite
graph, no random edges are needed at all.Comment: 21 pages; some typos fixed in the last versio
Recommended from our members
Combinatorics and Probability
The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers
On the Anti-Ramsey Threshold for Non-Balanced Graphs
For graphs G,H, we write Grb⟶H if for every proper edge-coloring of G there is a rainbow copy of H, i.e., a copy where no color appears more than once. Kohayakawa, Konstadinidis and the last author proved that the threshold for G(n,p)rb⟶H is at most n−1/m2(H). Previous results have matched the lower bound for this anti-Ramsey threshold for cycles and complete graphs with at least 5 vertices. Kohayakawa, Konstadinidis and the last author also presented an infinite family of graphs H for which the anti-Ramsey threshold is asymptotically smaller than n−1/m2(H). In this paper, we devise a framework that provides a richer family of such graphs
Recommended from our members
Combinatorics, Probability and Computing
The main theme of this workshop was the use of probabilistic
methods in combinatorics and theoretical computer science. Although
these methods have been around for decades, they are being refined all
the time: they are getting more and more sophisticated and powerful.
Another theme was the study of random combinatorial structures,
either for their own sake, or to tackle extremal questions. The workshop
also emphasized connections between probabilistic combinatorics and
discrete probability
Small rainbow cliques in randomly perturbed dense graphs
For two graphs and , write if has the property that every \emph{proper} colouring of its edges
yields a \emph{rainbow} copy of .
We study the thresholds for such so-called \emph{anti-Ramsey} properties in
randomly perturbed dense graphs, which are unions of the form , where is an -vertex graph with edge-density at least
, and is independent of .
In a companion article, we proved that the threshold for the property
is
, whenever . For smaller , the thresholds behave more erratically, and for
they deviate downwards significantly from the aforementioned
aesthetic form capturing the thresholds for \emph{large} cliques.
In particular, we show that the thresholds for are
, , and , respectively. For we
determine the threshold up to a -factor in the exponent: they are
and , respectively. For , the
threshold is ; this follows from a more general result about odd cycles
in our companion paper.Comment: 37 pages, several figures; update following reviewer(s) comment
Small rainbow cliques in randomly perturbed dense graphs
For two graphs G and H, write G
rbw
−→ H if G has the property that every proper colouring
of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey
properties in randomly perturbed dense graphs, which are unions of the form G ∪ G(n, p), where
G is an n-vertex graph with edge-density at least d > 0, and d is independent of n.
In a companion paper, we proved that the threshold for the property G ∪ G(n, p)
rbw
−→ K` is
n
−1/m2(Kd`/2e)
, whenever ` ≥ 9. For smaller `, the thresholds behave more erratically, and for
4 ≤ ` ≤ 7 they deviate downwards significantly from the aforementioned aesthetic form capturing
the thresholds for large cliques.
In particular, we show that the thresholds for ` ∈ {4, 5, 7} are n
−5/4
, n
−1
, and n
−7/15, respectively. For ` ∈ {6, 8} we determine the threshold up to a (1 + o(1))-factor in the exponent: they
are n
−(2/3+o(1)) and n
−(2/5+o(1)), respectively. For ` = 3, the threshold is n
−2
; this follows from
a more general result about odd cycles in our companion paper
Recommended from our members
Combinatorics and Probability
For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices
- …