On topological and geometric (194) configurations

An $\left(n_k\right)$ configuration is a set of  $n$ points and  $n$ lines such that each point lies on  $k$ lines while each line contains  $k$ points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of $\left(n_k\right)$ configurations for a given  $k$ has been subject to active research. A current front of research concerns geometric $\left(n_4\right)$ configurations: it is now known that geometric $\left(n_4\right)$ configurations exist for all  $n\ge 18$, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of $\left(19_4\right)$ configurations: we obtain all topological $\left(19_4\right)$ configurations among which none are geometrically realizable.     

Further results on (v,4,1)-perfect difference families

In this paper, it is shown that an ASP$(2r+1,3)$ exists for $2\le r\le 100$ and $r\ne 4,5$. The existence of a $(v,4,1)$-PDF is investigated by taking advantage of the relationship between ASPs and perfect difference families (PDFs). It is proved that a $(12t+1,4,1)$-PDF exists for $t\le 100$ and $t\ne 2,3$. Several recursive constructions for ASPs and PDFs are also presented. As a consequence, the existence results of an optimal $(v,4,1)$-OOC is updated.     

Tauberian theorems for the (J,p) summability method

In this paper, we prove Tauberian theorems of slowly oscillating type for the $(J,p)$ summability method by using a result due to Mikhalin [G.A. Mikhalin, Theorem of Tauberian type for (J, pn) summation methods, Ukrain. Mat. Zh. 29 (1977), 763–770. English translation: Ukrain. Math. J. 29 (1977) 564–569].     

Quantitative (p,q) theorems in combinatorial geometry

We show quantitative versions of classical results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of Bárány, the existence of weak epsilon-nets for convex sets and the $(p,q)$ theorem of Alon and Kleitman. These methods can be applied to functions such as the volume, surface area or number of points of a discrete set. We also give general quantitative versions of the colorful Helly theorem for continuous functions.     

RU and (U,N)-implications satisfying Modus Ponens

In this paper it is investigated when some kinds of fuzzy implication functions derived from uninorms satisfy the Modus Ponens with respect to a continuous t-norm T, or equivalently, when they are T-conditionals. The study is done for RU-implications and $(U,N)$-implications with N a continuous fuzzy negation leading to a lot of solutions in both cases. For RU-implications T-conditionality only depends on the underlying t-norm of the uninorm used to derive the residual implication. On the contrary, for $(U,N)$-implications the underlying t-norm is never relevant and only the region out of the t-norm is so. Even the t-conorm can be not relevant also in some cases.     

Structure and algorithms for (cap,even hole)-free graphs

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph $G$ has a vertex of degree at most $\frac{3}{2}\omega \left(G\right)?1$, and hence $\chi \left(G\right)\le \frac{3}{2}\omega \left(G\right)$, where $\omega \left(G\right)$ denotes the size of a largest clique in $G$ and $\chi \left(G\right)$ denotes the chromatic number of $G$. We give an $O\left(nm\right)$ algorithm for $q$-coloring these graphs for fixed $q$ and an $O\left(nm\right)$ algorithm for maximum weight stable set, where $n$ is the number of vertices and $m$ is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring.Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs $G$ without clique cutsets have treewidth at most $6\omega \left(G\right)?1$ and clique-width at most 48.