5,128 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
Lower Bounds on the van der Waerden Numbers: Randomized- and Deterministic-Constructive
The van der Waerden number W(k,2) is the smallest integer n such that every
2-coloring of 1 to n has a monochromatic arithmetic progression of length k.
The existence of such an n for any k is due to van der Waerden but known upper
bounds on W(k,2) are enormous. Much effort was put into developing lower bounds
on W(k,2). Most of these lower bound proofs employ the probabilistic method
often in combination with the Lov\'asz Local Lemma. While these proofs show the
existence of a 2-coloring that has no monochromatic arithmetic progression of
length k they provide no efficient algorithm to find such a coloring. These
kind of proofs are often informally called nonconstructive in contrast to
constructive proofs that provide an efficient algorithm.
This paper clarifies these notions and gives definitions for deterministic-
and randomized-constructive proofs as different types of constructive proofs.
We then survey the literature on lower bounds on W(k,2) in this light. We show
how known nonconstructive lower bound proofs based on the Lov\'asz Local Lemma
can be made randomized-constructive using the recent algorithms of Moser and
Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to
transform these proofs into deterministic-constructive proofs. We provide
greatly simplified and fully self-contained proofs and descriptions for these
algorithms
The Wonder of Colors and the Principle of Ariadne
The Principle of Ariadne, formulated in 1988 ago by Walter Carnielli
and Carlos Di Prisco and later published in 1993, is an infinitary principle that is independent of the Axiom of Choice in ZF, although it can be consistently added to
the remaining ZF axioms. The present paper surveys, and motivates, the foundational importance of the Principle of Ariadne
and proposes the Ariadne Game, showing that the Principle of Ariadne,
corresponds precisely
to a winning strategy for the Ariadne Game. Some relations to other
alternative. set-theoretical principles
are also briefly discussed
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