60 research outputs found
Avoiding small subgraphs in Achlioptas processes
For a fixed integer r, consider the following random process. At each round,
one is presented with r random edges from the edge set of the complete graph on
n vertices, and is asked to choose one of them. The selected edges are
collected into a graph, which thus grows at the rate of one edge per round.
This is a natural generalization of what is known in the literature as an
Achlioptas process (the original version has r=2), which has been studied by
many researchers, mainly in the context of delaying or accelerating the
appearance of the giant component.
In this paper, we investigate the small subgraph problem for Achlioptas
processes. That is, given a fixed graph H, we study whether there is an online
algorithm that substantially delays or accelerates a typical appearance of H,
compared to its threshold of appearance in the random graph G(n, M). It is easy
to see that one cannot accelerate the appearance of any fixed graph by more
than the constant factor r, so we concentrate on the task of avoiding H. We
determine thresholds for the avoidance of all cycles C_t, cliques K_t, and
complete bipartite graphs K_{t,t}, in every Achlioptas process with parameter r
>= 2.Comment: 43 pages; reorganized and shortene
The codegree threshold for 3-graphs with independent neighborhoods
Given a family of 3-graphs F, we define its codegree threshold coex(n, F) to be the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. Let F3,2 be the 3-graph on {a, b, c, d, e} with 3-edges abc, abd, abe, and cde. In this paper, we give two proofs that coex(n, {F3,2}) = 1 3 + o(1) n, the first by a direct combinatorial argument and the second via a flag algebra computation. Information extracted from the latter proof is then used to obtain a stability result, from which in turn we derive the exact codegree threshold for all sufficiently large n: coex(n, {F3,2}) = n/3 − 1 if n is congruent to 1 modulo 3, and n/3 otherwise. In addition we determine the set of codegree-extremal configurations for all sufficiently large n
Hypergraph Tur\'an Problems in -Norm
There are various different notions measuring extremality of hypergraphs. In
this survey we compare the recently introduced notion of the codegree squared
extremal function with the Tur\'an function, the minimum codegree threshold and
the uniform Tur\'an density.
The codegree squared sum of a -uniform hypergraph
is defined to be the sum of codegrees squared over all pairs of
vertices . In other words, this is the square of the -norm of the
codegree vector. We are interested in how large can be if we
require to be -free for some -uniform hypergraph . This maximum
value of over all -free -vertex -uniform
hypergraphs is called the codegree squared extremal function, which we
denote by . We systemically study the extremal codegree
squared sum of various -uniform hypergraphs using various proof techniques.
Some of our proofs rely on the flag algebra method while others use more
classical tools such as the stability method. In particular, we
(asymptotically) determine the codegree squared extremal numbers of matchings,
stars, paths, cycles, and , the -vertex hypergraph with edge set
.
Additionally, our paper has a survey format, as we state several conjectures
and give an overview of Tur\'an densities, minimum codegree thresholds and
codegree squared extremal numbers of popular hypergraphs. We intend to update
the arXiv version of this paper regularly.Comment: Invited survey for BCC 2022, comments are welcom
The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models
The codegree of a lattice polytope is the smallest integer dilate that
contains a lattice point in the relative interior. The weak maximum likelihood
threshold of a statistical model is the smallest number of data points for
which there is a non-zero probability that the maximum likelihood estimate
exists. The codegree of a marginal polytope is a lower bound on the maximum
likelihood threshold of the associated log-linear model, and they are equal
when the marginal polytope is normal. We prove a lower bound on the codegree in
the case of hierarchical log-linear models and provide a conjectural formula
for the codegree in general. As an application, we study when the marginal
polytopes of hierarchical models are Gorenstein, including a classification of
Gorenstein decomposable models, and a conjectural classification of Gorenstein
binary hierarchical models
When does the K_4-free process stop?
The K_4-free process starts with the empty graph on n vertices and at each
step adds a new edge chosen uniformly at random from all remaining edges that
do not complete a copy of K_4. Let G be the random maximal K_4-free graph
obtained at the end of the process. We show that for some positive constant C,
with high probability as , the maximum degree in G is at most . This resolves a conjecture of Bohman and Keevash for
the K_4-free process and improves on previous bounds obtained by Bollob\'as and
Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash
this shows that with high probability G has
edges and is `nearly regular', i.e., every vertex has degree
. This answers a question of Erd\H{o}s, Suen
and Winkler for the K_4-free process. We furthermore deduce an additional
structural property: we show that whp the independence number of G is at least
, which matches an upper bound
obtained by Bohman up to a factor of . Our analysis of the
K_4-free process also yields a new result in Ramsey theory: for a special case
of a well-studied function introduced by Erd\H{o}s and Rogers we slightly
improve the best known upper bound.Comment: 39 pages, 3 figures. Minor edits. To appear in Random Structures and
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