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A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cghs consisting of two disjoint triples. These were studied at length by Bra{ss} (2004) and by Aronov, Dujmovic, Morin, Ooms, and da Silveira (2019). We determine the extremal functions exactly for seven of the eight configurations. The above results are about cyclically ordered hypergraphs. We extend some of them for triangle systems with vertices from a non-convex set. We also solve problems posed by P. Frankl, Holmsen and Kupavskii (2020), in particular, we determine the exact maximum size of an intersecting family of triangles whose vertices come from a set of $n$ points in the plane.
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In a generalized Turan problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.
The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path in place of `geodesic. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers, with applications to a realisation result. We then solve a Tur{a}n problem for the size of graphs with given order and position numbers and characterise the possible diameters of graphs with given order and monophonic position number. Finally we classify the graphs with given order and diameter and largest possible general position number.
We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open problems, and formulate new ones.
A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from ${1, 2, ldots, k}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we consider two extremal problems related to Gallai-$k$-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a $k$-edge-coloring of $K_n$. Second, for $ngeq GR_k(K_3)$, we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-$k$-coloring of $K_{n}$, yielding the exact value for $k=3$. Furthermore, we determine the Gallai-Ramsey number $GR_k(K_4+e)$ for the graph on five vertices consisting of a $K_4$ with a pendant edge.
Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particular, we show that if a set $P$ of $n$ points is in convex position, then the largest intersecting family of triangles with vertices in $P$ contains at most $(frac{1}{4}+o(1))binom{n}{3}$ triangles.
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