33 research outputs found
On the lower tail variational problem for random graphs
We study the lower tail large deviation problem for subgraph counts in a
random graph. Let denote the number of copies of in an
Erd\H{o}s-R\'enyi random graph . We are interested in
estimating the lower tail probability for fixed .
Thanks to the results of Chatterjee, Dembo, and Varadhan, this large
deviation problem has been reduced to a natural variational problem over
graphons, at least for (and conjecturally for a larger
range of ). We study this variational problem and provide a partial
characterization of the so-called "replica symmetric" phase. Informally, our
main result says that for every , and for some
, as slowly, the main contribution to the lower tail
probability comes from Erd\H{o}s-R\'enyi random graphs with a uniformly tilted
edge density. On the other hand, this is false for non-bipartite and
close to 1.Comment: 15 pages, 5 figures, 1 tabl
Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their
properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic
and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization,
Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions.
This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session
Bounds on Ramsey Games via Alterations
This note contains a refined alteration approach for constructing H-free
graphs: we show that removing all edges in H-copies of the binomial random
graph does not significantly change the independence number (for suitable
edge-probabilities); previous alteration approaches of Erdos and Krivelevich
remove only a subset of these edges. We present two applications to online
graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper,
Scissors games and online Ramsey numbers.Comment: 9 page
Counting extensions revisited
We consider rooted subgraphs in random graphs, i.e., extension counts such as
(i) the number of triangles containing a given vertex or (ii) the number of
paths of length three connecting two given vertices. In 1989, Spencer gave
sufficient conditions for the event that, with high probability, these
extension counts are asymptotically equal for all choices of the root vertices.
For the important strictly balanced case, Spencer also raised the fundamental
question whether these conditions are necessary. We answer this question by a
careful second moment argument, and discuss some intriguing problems that
remain open.Comment: 21 pages, 2 figure