109 research outputs found
Monochromatic loose paths in multicolored -uniform cliques
For integers and , a -uniform hypergraph is called a
loose path of length , and denoted by , if it consists of
edges such that if and
if . In other words, each pair of
consecutive edges intersects on a single vertex, while all other pairs are
disjoint. Let be the minimum integer such that every
-edge-coloring of the complete -uniform hypergraph yields a
monochromatic copy of . In this paper we are mostly interested in
constructive upper bounds on , meaning that on the cost of
possibly enlarging the order of the complete hypergraph, we would like to
efficiently find a monochromatic copy of in every coloring. In
particular, we show that there is a constant such that for all ,
, , and , there is an
algorithm such that for every -edge-coloring of the edges of , it
finds a monochromatic copy of in time at most . We also
prove a non-constructive upper bound
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
Using the Incompressibility Method to obtain Local Lemma results for Ramsey-type Problems
We reveal a connection between the incompressibility method and the Lovasz
local lemma in the context of Ramsey theory. We obtain bounds by repeatedly
encoding objects of interest and thereby compressing strings. The method is
demonstrated on the example of van der Waerden numbers. It applies to lower
bounds of Ramsey numbers, large transitive subtournaments and other Ramsey
phenomena as well.Comment: 8 pages, 1 figur
Complexity of Token Swapping and its Variants
In the Token Swapping problem we are given a graph with a token placed on
each vertex. Each token has exactly one destination vertex, and we try to move
all the tokens to their destinations, using the minimum number of swaps, i.e.,
operations of exchanging the tokens on two adjacent vertices. As the main
result of this paper, we show that Token Swapping is -hard parameterized
by the length of a shortest sequence of swaps. In fact, we prove that, for
any computable function , it cannot be solved in time where is the number of vertices of the input graph, unless the ETH
fails. This lower bound almost matches the trivial -time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored
Token Swapping (where the tokens have different colors and tokens of the same
color are indistinguishable), and Subset Token Swapping (where each token has a
set of possible destinations). To complement the hardness result, we prove that
even the most general variant, Subset Token Swapping, is FPT in nowhere-dense
graph classes.
Finally, we consider the complexities of all three problems in very
restricted classes of graphs: graphs of bounded treewidth and diameter, stars,
cliques, and paths, trying to identify the borderlines between polynomial and
NP-hard cases.Comment: 23 pages, 7 Figure
Constructive Lower Bounds on Classical Multicolor Ramsey Numbers
This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for R_k(4) and R_k(5) for some small k, including 415 \u3c = R_3(5), 634 \u3c = R_4(4), 2721 \u3c = R_4(5), 3416 \u3c = R_5(4) and 26082 \u3c = R_5(5). Most of the new lower bounds are consequences of general constructions
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
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