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Register a new userThe poset of copies for automorphism groups of countable relational structures
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Let $mathrm{G}$ be a subgroup of the symmetric group $mathfrak S(U)$ of all permutations of a countable set $U$. Let $overline{mathrm{G}}$ be the topological closure of $mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $overline{mathrm{G}}[U]:={f[U]mid fin overline{mathrm{G}}}$ of images of the functions in $overline{mathrm{G}}$, being ordered under inclusion. This set $overline{mathrm{G}}[U]$ of subsets of the set $U$ will be called the emph{poset of copies for} the group $mathrm{G}$. A denomination being justified by the fact that for every subgroup $mathrm{G}$ of the symmetric group $mathfrak S(U)$ there exists a homogeneous relational structure $R$ on $U$ such that $overline G$ is the set of embeddings of the homogeneous structure $R$ into itself and $overline{mathrm{G}}[U]$ is the set of copies of $R$ in $R$ and that the set of bijections $overline Gcap mathfrak S(U)$ of $U$ to $U$ forms the group of automorphisms of $mathrm{R}$.
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The maximum size, $La(n,P)$, of a family of subsets of $[n]={1,2,...,n}$ without containing a copy of $P$ as a subposet, has been intensively studied. Let $P$ be a graded poset. We say that a family $mathcal{F}$ of subsets of $[n]={1,2,...,n}$ contains a emph{rank-preserving} copy of $P$ if it contains a copy of $P$ such that elements of $P$ having the same rank are mapped to sets of same size in $mathcal{F}$. The largest size of a family of subsets of $[n]={1,2,...,n}$ without containing a rank-preserving copy of $P$ as a subposet is denoted by $La_{rp}(n,P)$. Clearly, $La(n,P) le La_{rp}(n,P)$ holds. In this paper we prove asymptotically optimal upper bounds on $La_{rp}(n,P)$ for tree posets of height $2$ and monotone tree posets of height $3$, strengthening a result of Bukh in these cases. We also obtain the exact value of $La_{rp}(n,{Y_{h,s},Y_{h,s}})$ and $La(n,{Y_{h,s},Y_{h,s}})$, where $Y_{h,s}$ denotes the poset on $h+s$ elements $x_1,dots,x_h,y_1,dots,y_s$ with $x_1<dots<x_h<y_1,dots,y_s$ and $Y_{h,s}$ denotes the dual poset of $Y_{h,s}$.
Let $B_n$ be the poset generated by the subsets of $[n]$ with the inclusion as relation and let $P$ be a finite poset. We want to embed $P$ into $B_n$ as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for all finite posets $P$ as $frac{{n choose lfloor n/2rfloor}}{M(P)}$, where $M(P)$ denotes the minimal size of the convex hull of a copy of $P$. We discuss both weak and strong (induced) embeddings.
The distinguishing number of a graph $G$ is the smallest positive integer $r$ such that $G$ has a labeling of its vertices with $r$ labels for which there is no non-trivial automorphism of $G$ preserving these labels. Albertson and Collins computed the distinguishing number for various finite graphs, and Imrich, Klavv{z}ar and Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed graphs, and posets. We show that this number is in most cases two or infinite, and besides a few exceptions conjecture that this is so for all primitive homogeneous countable structures.
If $G$ is a free product of finite groups, let $Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] and Guti{e}rrez-Krsti{c} [M. Guti{e}rrez and S. Krsti{c}, {em Normal forms for the group of basis-conjugating automorphisms of a free group}, International Journal of Algebra and Computation 8 (1998) 631-669] derived (also see Bogley-Krsti{c} [W. Bogley and S. Krsti{c}, {em String groups and other subgroups of $Aut(F_n)$}, preprint] space of pointed trees is an $underline{E} Sigma Aut_1(G)$-space for these groups.
We introduce an extension, indexed by a partially ordered set P and cardinal numbers k,l, denoted by (k,l)-->P, of the classical relation (k,n,l)--> r in infinite combinatorics. By definition, (k,n,l)--> r holds, if every map from the n-element subsets of k to the subsets of k with less than l elements has a r-element free set. For example, Kuratowskis Free Set Theorem states that (k,n,l)-->n+1 holds iff k is larger than or equal to the n-th cardinal successor l^{+n} of the infinite cardinal k. By using the (k,l)-->P framework, we present a self-contained proof of the first authors result that (l^{+n},n,l)-->n+2, for each infinite cardinal l and each positive integer n, which solves a problem stated in the 1985 monograph of Erdos, Hajnal, Mate, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation (l^{+(n-1)},r,l)-->2^m, where m is the largest integer below (1/2)(1-2^{-r})^{-n/r}, for every infinite cardinal l and all positive integers n and r with r larger than 1 but smaller than n. For example, (aleph_{210},4,aleph_0)-->32,768. Other order-dimension estimates yield relations such as (aleph_{109},4,aleph_0)--> 257 (using an estimate by Furedi and Kahn) and (aleph_7,4,aleph_0)-->10 (using an exact estimate by Dushnik).
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