15 research outputs found
Induced Ramsey-type theorems
We present a unified approach to proving Ramsey-type theorems for graphs with
a forbidden induced subgraph which can be used to extend and improve the
earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham,
and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by
Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's
regularity lemma, thereby giving much better bounds. The same approach can be
also used to show that pseudo-random graphs have strong induced Ramsey
properties. This leads to explicit constructions for upper bounds on various
induced Ramsey numbers.Comment: 30 page
On the existence of highly organized communities in networks of locally interacting agents
In this paper we investigate phenomena of spontaneous emergence or purposeful
formation of highly organized structures in networks of related agents. We show
that the formation of large organized structures requires exponentially large,
in the size of the structures, networks. Our approach is based on Kolmogorov,
or descriptional, complexity of networks viewed as finite size strings. We
apply this approach to the study of the emergence or formation of simple
organized, hierarchical, structures based on Sierpinski Graphs and we prove a
Ramsey type theorem that bounds the number of vertices in Kolmogorov random
graphs that contain Sierpinski Graphs as subgraphs. Moreover, we show that
Sierpinski Graphs encompass close-knit relationships among their vertices that
facilitate fast spread and learning of information when agents in their
vertices are engaged in pairwise interactions modelled as two person games.
Finally, we generalize our findings for any organized structure with succinct
representations. Our work can be deployed, in particular, to study problems
related to the security of networks by identifying conditions which enable or
forbid the formation of sufficiently large insider subnetworks with malicious
common goal to overtake the network or cause disruption of its operation
Induced ramsey number for a star versus a fixed graph
We write F{\buildrel {\text{ind}} \over \longrightarrow}(H,G) for graphs F, G, and H, if for any coloring of the edges of F in red and blue, there is either a red induced copy of H or a blue induced copy of G. For graphs G and H, let IR(H, G) be the smallest number of vertices in a graph F such that F{\buildrel {\text{ind}} \over \longrightarrow}(H,G).
In this note we consider the case when G is a star on n edges, for large n and H is a fixed graph. We prove that
,
for any , sufficiently large n, and χ(H) denoting the chromatic number of H. The lower bound is asymptotically tight for any fixed bipartite H. The upper bound is attained up to a constant factor, for example when H is a clique
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
An efficient container lemma
We prove a new, efficient version of the hypergraph container theorems that
is suited for hypergraphs with large uniformities. The main novelty is a
refined approach to constructing containers that employs simple ideas from
high-dimensional convex geometry. The existence of smaller families of
containers for independent sets in such hypergraphs, which is guaranteed by the
new theorem, allows us to improve upon the best currently known bounds for
several problems in extremal graph theory, discrete geometry, and Ramsey
theory.Comment: 56 pages, revised versio