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In this paper, we study product-free subsets of the free semigroup over a finite alphabet $A$. We prove that the maximum density of a product-free subset of the free semigroup over $A$, with respect to the natural measure that assigns a weight of $|A|^{-n}$ to each word of length $n$, is precisely $1/2$.
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A subset $U$ of a set $S$ with a binary operation is called {it avoidable} if $S$ can be partitioned into two subsets $A$ and $B$ such that no element of $U$ can be written as a product of two distinct elements of $A$ or as the product of two distinct elements of $B$. The avoidable sets of the bicyclic inverse semigroup are classified.
We investigate the relationship between two constructions of maximal comma-free codes described respectively by Eastman and by Scholtz and the notions of Hall sets and Lazard sets introduced in connection with factorizations of free monoids and bases of free Lie algebras.
Nielsen proved that the maximum number of maximal independent sets (MISs) of size $k$ in an $n$-vertex graph is asymptotic to $(n/k)^k$, with the extremal construction a disjoint union of $k$ cliques with sizes as close to $n/k$ as possible. In this paper we study how many MISs of size $k$ an $n$-vertex graph $G$ can have if $G$ does not contain a clique $K_t$. We prove for all fixed $k$ and $t$ that there exist such graphs with $n^{lfloorfrac{(t-2)k}{t-1}rfloor-o(1)}$ MISs of size $k$ by utilizing recent work of Gowers and B. Janzer on a generalization of the Ruzsa-Szemeredi problem. We prove that this bound is essentially best possible for triangle-free graphs when $kle 4$.
This paper studies the maximal size of product-free sets in Z/nZ. These are sets of residues for which there is no solution to ab == c (mod n) with a,b,c in the set. In a previous paper we constructed an infinite sequence of integers (n_i)_{i > 0} and product-free sets S_i in Z/n_iZ such that the density |S_i|/n_i tends to 1 as i tends to infinity, where |S_i|$ denotes the cardinality of S_i. Here we obtain matching, up to constants, upper and lower bounds on the maximal attainable density as n tends to infinity.
We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
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