Let $u_k(G,p)$ be the maximum over all $k$-vertex graphs $F$ of by how much the number of induced copies of $F$ in $G$ differs from its expectation in the Erd\H{o}s-R\'enyi random graph with the same number of vertices as $G$ and with edge probability $p$. This may be viewed as a measure of how close $G$ is to being $p$-quasirandom. For a positive integer $n$ and $0
Falgas-Ravry, Victor;
Marchant, Edward;
Pikhurko, Oleg;
Vaughan, Emil R. (2015)Projects:
EC | EC (306493) Given a family of 3-graphs F, we define its codegree threshold coex(n, F) to be the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. Let F3,2 be the 3-graph on {a, b, c, d, e} with 3-edges abc, abd, abe, and cde. In this paper, we give two proofs that coex(n, {F3,2}) = 1 3 + o(1) n, the first by a direct combinatorial argument and the second via a flag algebra ...
Grabowski, Łukasz;
Máthé, András;
Pikhurko, Oleg (2014)Projects:
EC | EC (306493) We give a sketch of proof that any two (Lebesgue) measurable subsets of the unit sphere in $R^n$, for $n\ge 3$, with non-empty interiors and of the same measure are equidecomposable using pieces that are measurable.
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.
Falgas-Ravry, Victor;
Marchant, Edward;
Pikhurko, Oleg;
Vaughan, Emil (2013)Projects:
EC | EC (306493) Given a family of 3-graphs $F$, we define its codegree threshold $\mathrm{coex}(n, F)$ to be the largest number $d=d(n)$ such that there exists an $n$-vertex 3-graph in which every pair of vertices is contained in at least $d$ 3-edges but which contains no member of $F$ as a subgraph. Let $F_{3,2}$ be the 3-graph on $\{a,b,c,d,e\}$ with 3-edges $\{abc,abd,abe,cde\}$. In this paper, we give two proofs that $\mathrm{coex}(n, F_{3,2})= n/3 +o(n)$, the first by a direct combinatorial argument and...
Pikhurko, Oleg;
Staden, Katherine;
Yilma, Zelealem B. (2016)Projects:
EC | EC (306493) Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it has been solved only for a ...
The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by K\H{o}nig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge-colorings for graphings. A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence theory and measurable group theory. We show that ever...
In this note, we prove that the base case of the Graham--Rothschild Theorem, i.e., the one that considers colorings of the ($1$-dimensional) variable words, admits bounds in the class $\mathcal{E}^5$ of Grzegorczyk's hierarchy.
Let $G$ be a graph whose edges are coloured with $k$ colours, and $\mathcal H=(H_1,\dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $\mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1\le i\le k$. Let $\phi_{k}(n,\mathcal H)$ be the smallest number $\phi$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $\mathcal H$...
The Tur\'an density \pi(H) of a family H of k-graphs is the limit as n tends to infinity of the maximum edge density of an H-free k-graph on n vertices. Let I^k consist of all possible Tur\'an densities and let F^k be the set of Tur\'an densities of finite k-graph families. Here we prove that F^k contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that F^k contains an...