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Milos S. Kurilic

Publications and source records attributed to Milos S. Kurilic.

6 recordsLinked to original sources
arXiv2013

Maximal Chains of Isomorphic Subgraphs of Countable Ultrahomogeneous Graphs

For a countable ultrahomogeneous graph G let P(G) denote the collection of domains of subgraphs of G isomorphic to G. The order types of maximal chains in the set P(G) U ø ordered by the inclusion are characterized as: (I) the order types of compact sets of reals having the minimum non-isolated, if G is the Rado graph or the Henson graph H_n, for some n>2; (II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of μdisjoint complete graphs of size ν, where μν=ω.

math.LO↗
arXiv2013

Four Games on Boolean Algebras

The games G_2 and G_3 are played on a complete Boolean algebra B in ω-many moves. At the beginning White picks a non-zero element p of B and, in the n-th move, White picks a positive p_n < p and Black chooses an i_n belonging to {0,1}. White wins G_2 iff liminf p_n^{i_n}=0 and wins G_3 iff \bigvee_{A\in [ω]^ω}\bigwedge_{n\in A}p_n^{i_n}=0. It is shown that White has a winning strategy in the game G_2 iff White has a winning strategy in the cut-and-choose game G_c&c introduced by Jech. Also, White has a winning strategy in the game G_3 iff forcing by B produces a subset R of the binary tree 2^{<ω} containing either f^0 or f^1, for each f in 2^{<ω}, and having unsupported intersection with each branch of the tree 2^{<ω} belonging to V. On the other hand, if forcing by B produces independent (splitting) reals then White has a winning strategy in the game G_3 played on B. It is shown that $\diamondsuit$ implies the existence of an algebra on which these games are undetermined.

math.LO↗
arXiv2013

Maximal Chains of Isomorphic Suborders of Countable Ultrahomogeneous Partial Orders

We investigate the poset (P(X),\subset), where P(X) is the set of isomorphic suborders of a countable ultrahomogeneous partial order X. For X different from (resp. equal to) a countable antichain the order types of maximal chains in (P(X)\cup \{\emptyset \},\subset) are characterized as the order types of compact (resp. compact and nowhere dense) sets of reals having the minimum non-isolated.

math.LO↗
arXiv2013

Maximally Embeddable Components

We investigate the partial orderings of the form (P(X),\subset), where X is a countable binary relational structure and P(X) the set of the domains of its isomorphic substructures and show that if the components of X are maximally embeddable and satisfy an additional condition related to connectivity, then the poset (P(X),\subset) is forcing equivalent to a finite power of (P(ω)/Fin)^+, or to (P(ω\times ω)/(Fin \times Fin))^+, or to the direct product (P(Δ)/ED_fin)^+ \times ((P(ω)/Fin)^+)^n, for some n \in ω. In particular we obtain forcing equivalents of the posets of copies of countable equivalence relations, disconnected ultrahomogeneous graphs and some partial orderings.

math.LO↗
arXiv2013

Posets of Copies of Countable Scattered Linear Orders

We show that the separative quotient of the poset (P(L),\subset) of isomorphic suborders of a countable scattered linear order L is σ-closed and atomless. So, under the CH, all these posets are forcing-equivalent (to P(ω)/Fin).

math.LO↗