1,605 research outputs found
A counterexample to Thiagarajan's conjecture on regular event structures
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002)
that regular event structures correspond exactly to event structures obtained
as unfoldings of finite 1-safe Petri nets. The same counterexample is used to
disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999)
that domains of regular event structures with bounded -cliques are
recognizable by finite trace automata. Event structures, trace automata, and
Petri nets are fundamental models in concurrency theory. There exist nice
interpretations of these structures as combinatorial and geometric objects.
Namely, from a graph theoretical point of view, the domains of prime event
structures correspond exactly to median graphs; from a geometric point of view,
these domains are in bijection with CAT(0) cube complexes.
A necessary condition for both conjectures to be true is that domains of
regular event structures (with bounded -cliques) admit a regular nice
labeling. To disprove these conjectures, we describe a regular event domain
(with bounded -cliques) that does not admit a regular nice labeling.
Our counterexample is derived from an example by Wise (1996 and 2007) of a
nonpositively curved square complex whose universal cover is a CAT(0) square
complex containing a particular plane with an aperiodic tiling. We prove that
other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way
deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009).
On the positive side, using breakthrough results by Agol (2013) and Haglund
and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's
conjecture is true for regular event structures whose domains occur as
principal filters of hyperbolic CAT(0) cube complexes which are universal
covers of finite nonpositively curved cube complexes
Ramsey numbers of cubes versus cliques
The cube graph Q_n is the skeleton of the n-dimensional cube. It is an
n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N
such that every graph of order N contains the cube graph Q_n or an independent
set of order s. Burr and Erdos in 1983 asked whether the simple lower bound
r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large.
We make progress on this problem, obtaining the first upper bound which is
within a constant factor of the lower bound.Comment: 26 page
Alternating sign matrices and domino tilings
We introduce a family of planar regions, called Aztec diamonds, and study the
ways in which these regions can be tiled by dominoes. Our main result is a
generating function that not only gives the number of domino tilings of the
Aztec diamond of order but also provides information about the orientation
of the dominoes (vertical versus horizontal) and the accessibility of one
tiling from another by means of local modifications. Several proofs of the
formula are given. The problem turns out to have connections with the
alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square
ice model studied by Lieb
Coxeter groups and random groups
For every dimension d, there is an infinite family of convex co-compact
reflection groups of isometries of hyperbolic d-space --- the superideal
(simplicial and cubical) reflection groups --- with the property that a random
group at any density less than a half (or in the few relators model) contains
quasiconvex subgroups commensurable with some member of the family, with
overwhelming probability.Comment: 18 pages, 14 figures; version 2 incorporates referee's correction
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