669 research outputs found
On Martin's Pointed Tree Theorem
We investigate the reverse mathematics strength of Martin's pointed tree
theorem (MPT) and one of its variants, weak Martin's pointed tree theorem
(wMPT)
A new approach to nonrepetitive sequences
A sequence is nonrepetitive if it does not contain two adjacent identical
blocks. The remarkable construction of Thue asserts that 3 symbols are enough
to build an arbitrarily long nonrepetitive sequence. It is still not settled
whether the following extension holds: for every sequence of 3-element sets
there exists a nonrepetitive sequence with
. Applying the probabilistic method one can prove that this is true
for sufficiently large sets . We present an elementary proof that sets of
size 4 suffice (confirming the best known bound). The argument is a simple
counting with Catalan numbers involved. Our approach is inspired by a new
algorithmic proof of the Lov\'{a}sz Local Lemma due to Moser and Tardos and its
interpretations by Fortnow and Tao. The presented method has further
applications to nonrepetitive games and nonrepetitive colorings of graphs.Comment: 5 pages, no figures.arXiv admin note: substantial text overlap with
arXiv:1103.381
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