37 research outputs found
Ramsey-type constructions for arrangements of segments
Improving a result of K\'arolyi, Pach and T\'oth, we construct an arrangement
of segments in the plane with at most pairwise
crossing or pairwise disjoint segments. We use the recursive method based on
flattenable arrangements which was established by Larman, Matou\v{s}ek, Pach
and T\"or\H{o}csik. We also show that not every arrangement can be flattened,
by constructing an intersection graph of segments which cannot be realized by
an arrangement of segments crossing a common line. Moreover, we also construct
an intersection graph of segments crossing a common line which cannot be
realized by a flattenable arrangement.Comment: 11 pages, 6 figure
Density theorems for intersection graphs of t-monotone curves
A curve \gamma in the plane is t-monotone if its interior has at most t-1
vertical tangent points. A family of t-monotone curves F is \emph{simple} if
any two members intersect at most once. It is shown that if F is a simple
family of n t-monotone curves with at least \epsilon n^2 intersecting pairs
(disjoint pairs), then there exists two subfamilies F_1,F_2 \subset F of size
\delta n each, such that every curve in F_1 intersects (is disjoint to) every
curve in F_2, where \delta depends only on \epsilon. We apply these results to
find pairwise disjoint edges in simple topological graphs
Erdos-Hajnal-type theorems in hypergraphs
The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free,
that is, it does not contain an induced copy of a given graph H, then it must
contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0
depends only on the graph H. Except for a few special cases, this conjecture
remains wide open. However, it is known that a H-free graph must contain a
complete or empty bipartite graph with parts of polynomial size. We prove an
analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform
hypergraph on n vertices is H-free, for any given H, then it must contain a
complete or empty tripartite subgraph with parts of order c(log n)^{1/2 +
d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log
n)^{1/2}, which holds in all 3-uniform hypergraphs, and, up to the value of the
constant d(H), is best possible. We also prove that, for k > 3, no analogue of
the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That
is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which
do not contain cliques or independent sets of size appreciably larger than one
would normally expect.Comment: 15 page
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
The number of edges in k-quasi-planar graphs
A graph drawn in the plane is called k-quasi-planar if it does not contain k
pairwise crossing edges. It has been conjectured for a long time that for every
fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices
is O(n). The best known upper bound is n(\log n)^{O(\log k)}. In the present
note, we improve this bound to (n\log n)2^{\alpha^{c_k}(n)} in the special case
where the graph is drawn in such a way that every pair of edges meet at most
once. Here \alpha(n) denotes the (extremely slowly growing) inverse of the
Ackermann function. We also make further progress on the conjecture for
k-quasi-planar graphs in which every edge is drawn as an x-monotone curve.
Extending some ideas of Valtr, we prove that the maximum number of edges of
such graphs is at most 2^{ck^6}n\log n.Comment: arXiv admin note: substantial text overlap with arXiv:1106.095
On the number of touching pairs in a set of planar curves
Given a set of planar curves (Jordan arcs), each pair of which meets --
either crosses or touches -- exactly once, we establish an upper bound on the
number of touchings. We show that such a curve family has touchings,
where is the number of faces in the curve arrangement that contains at
least one endpoint of one of the curves. Our method relies on finding special
subsets of curves called quasi-grids in curve families; this gives some
structural insight into curve families with a high number of touchings.Comment: 14 pages, 7 figure