First order properties on nowhere dense structures

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special cl...

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On nowhere dense graphs

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special cl...

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Dimension , matroids , and dense pairs of first - order structures

A structure M is pregeometric if the algebraic closure is a pregeometry in all M ′ elementarily equivalent to M . We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique exi...

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Dimensions, matroids, and dense pairs of first-order structures

A structure M is pregeometric if the algebraic closure is a pregeometry in all M ′ elementarily equivalent to M . We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique exi...

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Reconfiguration on nowhere dense graph classes

In the token jumping problem for a vertex subset problem Q on graphs we are given a graph G and two feasible solutions Ss, St ⊆ V (G) of Q with |Ss| = |St|, and imagine that a token is placed on each vertex of Ss. The problem is to determine whether there exists a sequence S1, . . . , Sn of feasible solutions, where S1 = Ss, Sn = St and each Si+1 results from Si, 1 ≤ i < n, by moving exactly on...

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First order properties on nowhere dense structures

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