56 research outputs found
New Ramsey Classes from Old
Let C_1 and C_2 be strong amalgamation classes of finite structures, with
disjoint finite signatures sigma and tau. Then C_1 wedge C_2 denotes the class
of all finite (sigma cup tau)-structures whose sigma-reduct is from C_1 and
whose tau-reduct is from C_2. We prove that when C_1 and C_2 are Ramsey, then
C_1 wedge C_2 is also Ramsey. We also discuss variations of this statement, and
give several examples of new Ramsey classes derived from those general results.Comment: 11 pages. In the second version, to be submitted for journal
publication, a number of typos has been removed, and a grant acknowledgement
has been adde
Binary simple homogeneous structures are supersimple with finite rank
Suppose that M is an infinite structure with finite relational vocabulary
such that every relation symbol has arity at most 2. If M is simple and
homogeneous then its complete theory is supersimple with finite SU-rank which
cannot exceed the number of complete 2-types over the empty set
The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems
We prove that an -categorical core structure primitively positively
interprets all finite structures with parameters if and only if some stabilizer
of its polymorphism clone has a homomorphism to the clone of projections, and
that this happens if and only if its polymorphism clone does not contain
operations , , satisfying the identity .
This establishes an algebraic criterion equivalent to the conjectured
borderline between P and NP-complete CSPs over reducts of finitely bounded
homogenous structures, and accomplishes one of the steps of a proposed strategy
for reducing the infinite domain CSP dichotomy conjecture to the finite case.
Our theorem is also of independent mathematical interest, characterizing a
topological property of any -categorical core structure (the existence
of a continuous homomorphism of a stabilizer of its polymorphism clone to the
projections) in purely algebraic terms (the failure of an identity as above).Comment: 15 page
Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width
Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ? := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking.
Our main result is that the structures ? under consideration have relational width exactly (2, ?_?) where ?_? is the maximal size of a forbidden subgraph of ?, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_? may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3)
Canonical functions: a proof via topological dynamics
Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent algebraic characterisations of canonicity
Permutations on the random permutation
The random permutation is the Fra\"iss\'e limit of the class of finite
structures with two linear orders. Answering a problem stated by Peter Cameron
in 2002, we use a recent Ramsey-theoretic technique to show that there exist
precisely 39 closed supergroups of the automorphism group of the random
permutation, and thereby expose all symmetries of this structure. Equivalently,
we classify all structures which have a first-order definition in the random
permutation.Comment: 18 page
Generalized Indiscernibles as Model-complete Theories
We give an almost entirely model-theoretic account of both Ramsey classes of
finite structures and of generalized indiscernibles as studied in special cases
in (for example) [7], [9]. We understand "theories of indiscernibles" to be
special kinds of companionable theories of finite structures, and much of the
work in our arguments is carried in the context of the model-companion. Among
other things, this approach allows us to prove that the companion of a theory
of indiscernibles whose "base" consists of the quantifier-free formulas is
necessarily the theory of the Fraisse limit of a Fraisse class of linearly
ordered finite structures (where the linear order will be at least
quantifier-free definable). We also provide streamlined arguments for the
result of [6] identifying extremely amenable groups with the automorphism
groups of limits of Ramsey classes.Comment: 21 page
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