123 research outputs found
A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing
Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for
semi-algebraic -uniform hypergraphs of bounded complexity, showing that for
each the vertex set can be equitably partitioned into a bounded
number of parts (in terms of and the complexity) so that all but an
-fraction of the -tuples of parts are homogeneous. We prove that
the number of parts can be taken to be polynomial in . Our improved
regularity lemma can be applied to geometric problems and to the following
general question on property testing: is it possible to decide, with query
complexity polynomial in the reciprocal of the approximation parameter, whether
a hypergraph has a given hereditary property? We give an affirmative answer for
testing typical hereditary properties for semi-algebraic hypergraphs of bounded
complexity
Ramsey properties of algebraic graphs and hypergraphs
One of the central questions in Ramsey theory asks how small can be the size
of the largest clique and independent set in a graph on vertices. By the
celebrated result of Erd\H{o}s from 1947, the random graph on vertices with
edge probability , contains no clique or independent set larger than
, with high probability. Finding explicit constructions of graphs
with similar Ramsey-type properties is a famous open problem. A natural
approach is to construct such graphs using algebraic tools. Say that an
-uniform hypergraph is \emph{algebraic of complexity
} if the vertices of are elements of
for some field , and there exist polynomials
of degree at most
such that the edges of are determined by the zero-patterns of
. The aim of this paper is to show that if an algebraic graph
(or hypergraph) of complexity has good Ramsey properties, then at
least one of the parameters must be large. In 2001, R\'onyai, Babai and
Ganapathy considered the bipartite variant of the Ramsey problem and proved
that if is an algebraic graph of complexity on vertices, then
either or its complement contains a complete balanced bipartite graph of
size . We extend this result by showing that such
contains either a clique or an independent set of size
and prove similar results for algebraic hypergraphs of constant complexity. We
also obtain a polynomial regularity lemma for -uniform algebraic hypergraphs
that are defined by a single polynomial, that might be of independent interest.
Our proofs combine algebraic, geometric and combinatorial tools.Comment: 23 page
Semi-algebraic colorings of complete graphs
We consider -colorings of the edges of a complete graph, where each color
class is defined semi-algebraically with bounded complexity. The case
was first studied by Alon et al., who applied this framework to obtain
surprisingly strong Ramsey-type results for intersection graphs of geometric
objects and for other graphs arising in computational geometry. Considering
larger values of is relevant, e.g., to problems concerning the number of
distinct distances determined by a point set.
For and , the classical Ramsey number is the
smallest positive integer such that any -coloring of the edges of ,
the complete graph on vertices, contains a monochromatic . It is a
longstanding open problem that goes back to Schur (1916) to decide whether
, for a fixed . We prove that this is true if each color
class is defined semi-algebraically with bounded complexity. The order of
magnitude of this bound is tight. Our proof is based on the Cutting Lemma of
Chazelle {\em et al.}, and on a Szemer\'edi-type regularity lemma for
multicolored semi-algebraic graphs, which is of independent interest. The same
technique is used to address the semi-algebraic variant of a more general
Ramsey-type problem of Erd\H{o}s and Shelah
Bounded VC-Dimension Implies the Schur-Erd?s Conjecture
In 1916, Schur introduced the Ramsey number r(3;m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph K_n, there is a monochromatic copy of K?. He showed that r(3;m) ? O(m!), and a simple construction demonstrates that r(3;m) ? 2^?(m). An old conjecture of Erd?s states that r(3;m) = 2^?(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension
Erdős-Hajnal Conjecture for Graphs with Bounded VC-Dimension
The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(logn)1-o(1). The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, eclogn, due to Erds and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erds-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e(logn). Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovasz-Szegedy and Alon-Fischer-Newman, which is called the ultra-strong regularity lemma for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/epsilon)O(d), improving the original bound of (1/epsilon)O(d2) in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(nk)-time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey-Turan numbers for graphs with bounded VC-dimension
- …