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Abstract separation systems are a new unifying framework in which separations of graph, matroids and other combinatorial structures can be expressed and studied. We characterize the abstract separation systems that have representations as separation systems of graphs, sets, or set bipartitions.
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In this paper we enumerate the number of ways of selecting $k$ objects from $n$ objects arrayed in a line such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects and provide three different formulas when $m,pgeq 1$ and $ngeq pm(k-1)$. Also, we prove that the number of ways of selecting $k$ objects from $n$ objects arrayed in a circle such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects is given by $frac{n}{n-pk}binom{n-pk}{k}$, where $m,pgeq 1$ and $ngeq mpk+1$.
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.
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ${omega+1}$. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space.
In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Gra{ss}mannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a $2$-$(6,3,78)_5$ design by computer, which corresponds to a halving $operatorname{LS}_5[2](2,3,6)$. The application of the new recursion method to this halving and an already known $operatorname{LS}_3[2](2,3,6)$ yields two infinite two-parameter series of halvings $operatorname{LS}_3[2](2,k,v)$ and $operatorname{LS}_5[2](2,k,v)$ with integers $vgeq 6$, $vequiv 2mod 4$ and $3leq kleq v-3$, $kequiv 3mod 4$. Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$.
In this paper, we give a new lifting construction of hyperbolic type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to $m$-ovoids and $i$-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.
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