49 research outputs found

    New Ramsey Classes from Old

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    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

    Dense subsets of products of finite trees

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    We prove a "uniform" version of the finite density Halpern-L\"{a}uchli Theorem. Specifically, we say that a tree TT is homogeneous if it is uniquely rooted and there is an integer b≥2b\geq 2, called the branching number of TT, such that every t∈Tt\in T has exactly bb immediate successors. We show the following. For every integer d≥1d\geq 1, every b1,...,bd∈Nb_1,...,b_d\in\mathbb{N} with bi≥2b_i\geq 2 for all i∈{1,...,d}i\in\{1,...,d\}, every integer k\meg 1 and every real 0<ϵ≤10<\epsilon\leq 1 there exists an integer NN with the following property. If (T1,...,Td)(T_1,...,T_d) are homogeneous trees such that the branching number of TiT_i is bib_i for all i∈{1,...,d}i\in\{1,...,d\}, LL is a finite subset of N\mathbb{N} of cardinality at least NN and DD is a subset of the level product of (T1,...,Td)(T_1,...,T_d) satisfying ∣D∩(T1(n)×...×Td(n))∣≥ϵ∣T1(n)×...×Td(n)∣|D\cap \big(T_1(n)\times ...\times T_d(n)\big)| \geq \epsilon |T_1(n)\times ...\times T_d(n)| for every n∈Ln\in L, then there exist strong subtrees (S1,...,Sd)(S_1,...,S_d) of (T1,...,Td)(T_1,...,T_d) of height kk and with common level set such that the level product of (S1,...,Sd)(S_1,...,S_d) is contained in DD. The least integer NN with this property will be denoted by UDHL(b1,...,bd∣k,ϵ)UDHL(b_1,...,b_d|k,\epsilon). The main point is that the result is independent of the position of the finite set LL. The proof is based on a density increment strategy and gives explicit upper bounds for the numbers UDHL(b1,...,bd∣k,ϵ)UDHL(b_1,...,b_d|k,\epsilon).Comment: 36 pages, no figures; International Mathematics Research Notices, to appea

    Ramsey families of subtrees of the dyadic tree

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    Symmetries and Ramsey properties of trees

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    AbstractIn this paper we show the extent to which a finite tree of fixed height is a Ramsey object in the class of trees of the same height can be measured by its symmetry group
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