On kk-Gons and kk-Holes in Point Sets

We consider a variation of the classical Erd\H{o}s-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position

Problems and memories

I state some open problems coming from joint work with Paul Erd\H{o}sComment: This is a paper form of the talk I gave on July 5, 2013 at the centennial conference in Budapest to honor Paul Erd\H{o}

Subgraphs and Colourability of Locatable Graphs

We study a game of pursuit and evasion introduced by Seager in 2012, in which a cop searches the robber from outside the graph, using distance queries. A graph on which the cop wins is called locatable. In her original paper, Seager asked whether there exists a characterisation of the graph property of locatable graphs by either forbidden or forbidden induced subgraphs, both of which we answer in the negative. We then proceed to show that such a characterisation does exist for graphs of diameter at most 2, stating it explicitly, and note that this is not true for higher diameter. Exploring a different direction of topic, we also start research in the direction of colourability of locatable graphs, we also show that every locatable graph is 4-colourable, but not necessarily 3-colourable.Comment: 25 page

The anti-Ramsey threshold of complete graphs

For graphs $G$ and $H$, let G {\displaystyle\smash{\begin{subarray}{c} \hbox{\tiny\rm rb} \\ \longrightarrow \\ \hbox{\tiny\rm p} \end{subarray}}}H denote the property that for every proper edge-colouring of $G$ there is a rainbow $H$ in $G$. It is known that, for every graph $H$, an asymptotic upper bound for the threshold function $p^{\rm rb}_H=p^{\rm rb}_H(n)$ of this property for the random graph $G(n,p)$ is $n^{-1/m^{(2)}(H)}$, where $m^{(2)}(H)$ denotes the so-called maximum $2$-density of $H$. Extending a result of Nenadov, Person, \v{S}kori\'c, and Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower bound for $p^{\rm rb}_{K_k}$ for $k\geq 5$. Furthermore, we show that $p^{\rm rb}_{K_4} = n^{-7/15}$.Comment: 19 page

Asymptotic Structure of Graphs with the Minimum Number of Triangles

We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.Comment: 22 pages; 2 figure

Polytopes from Subgraph Statistics

Polytopes from subgraph statistics are important in applications and conjectures and theorems in extremal graph theory can be stated as properties of them. We have studied them with a view towards applications by inscribing large explicit polytopes and semi-algebraic sets when the facet descriptions are intractable. The semi-algebraic sets called curvy zonotopes are introduced and studied using graph limits. From both volume calculations and algebraic descriptions we find several interesting conjectures.Comment: Full article, 21 pages, 8 figures. Minor expository update