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We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that, under the same condition, there are four lines intersecting all the sets. In fact, we prove a colorful version of this result, under weakened conditions on the sets. A triple of sets $A,B,C$ in the plane is said to be a {em tight} if $textrm{conv}(Acup B)cap textrm{conv}(Acup C)cap textrm{conv}(Bcap C) eq emptyset.$ This notion was first introduced by Holmsen, where he showed that if $mathcal{F}$ is a family of compact convex sets in the plane in which every three sets form a tight triple, then there is a line intersecting at least $frac{1}{8}|mathcal{F}|$ members of $mathcal{F}$. Here we prove that if $mathcal{F}_1,dots,mathcal{F}_6$ are families of compact connected sets in the plane such that every three sets, chosen from three distinct families $mathcal{F}_i$, form a tight triple, then there exists $1le jle 6$ and three lines intersecting every member of $mathcal{F}_j$. In particular, this improves $frac{1}{8}$ to $frac{1}{3}$ in Holmsens result.
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Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words defined by the property of the extension graph of each word in the set to be acyclic or connected or a tree. We prove that in a uniformly recurrent tree set, the sets of first return words are bases of the free group on the alphabet. Concerning acyclic sets, we prove as a main result that a set $F$ is acyclic if and only if any bifix code included in $F$ is a basis of the subgroup that it generates.
A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $ngeq 2k + csqrt{kln k}$, almost all $k$-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $ngeq 2k+ 100ln k$. Our proof uses, among others, Sapozhenkos graph container lemma and the Das-Tran removal lemma.
A $k$-connected set in an infinite graph, where $k > 0$ is an integer, is a set of vertices such that any two of its subsets of the same size $ell leq k$ can be connected by $ell$ disjoint paths in the whole graph. We characterise the existence of $k$-connected sets of arbitrary but fixed infinite cardinality via the existence of certain minors and topological minors. We also prove a duality theorem for the existence of such $k$-connected sets: if a graph contains no such $k$-connected set, then it has a tree-decomposition which, whenever it exists, precludes the existence of such a $k$-connected set.
In finite group theory, studying the prime graph of a group has been an important topic for almost the past half-century. Recently, prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs without any knowledge of the group theoretical background. In this paper we study prime graphs from a linear algebra angle and focus on the class of minimally connected prime graphs introduced in earlier work on the subject. As our main results, we determine the determinants of the adjacency matrices and the spectra of some important families of these graphs.
For positive integers $n,r,k$ with $nge r$ and $kge2$, a set ${(x_1,y_1),(x_2,y_2),dots,(x_r,y_r)}$ is called a $k$-signed $r$-set on $[n]$ if $x_1,dots,x_r$ are distinct elements of $[n]$ and $y_1dots,y_rin[k]$. We say a $t$-intersecting family consisting of $k$-signed $r$-sets on $[n]$ is trivial if each member of this family contains a fixed $k$-signed $t$-set. In this paper, we determine the structure of large maximal non-trivial $t$-intersecting families. In particular, we characterize the non-trivial $t$-intersecting families with maximum size for $tge2$, extending a Hilton-Milner-type result for signed sets given by Borg.
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