Embedding tetrahedra into quasirandom hypergraphs

We investigate extremal problems for quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is $(d,\eta)$-quasirandom if for any subset $X\subseteq V$ and every set of pairs $P\subseteq V\times V$ the number of pairs $(x,(y,z))\in X\times P$ with $\{x,y,z\}$ being a hyperedge of $H$ is in the interval $d|X||P|\pm\eta|V|^3$. We show that for any $\varepsilon>0$ there exists $\eta>0$ such that every sufficiently large $(1/2+\varepsilon,\eta)$-quasirandom hypergraph contains a tetrahedron, i.e., four vertices spanning all four hyperedges. A known random construction shows that the density $1/2$ is best possible. This result is closely related to a question of Erd\H{o}s, whether every weakly quasirandom $3$-uniform hypergraph $H$ with density bigger than $1/2$, i.e., every large subset of vertices induces a hypergraph with density bigger than $1/2$, contains a tetrahedron.Comment: 18 pages, second version addresses changes arising from the referee report

An extremal problem in proper (r,p)(r,p)-coloring of hypergraphs

Let $G(V,E)$ be a $k$-uniform hypergraph. A hyperedge $e \in E$ is said to be properly $(r,p)$ colored by an $r$-coloring of vertices in $V$ if $e$ contains vertices of at least $p$ distinct colors in the $r$-coloring. An $r$-coloring of vertices in $V$ is called a {\it strong $(r,p)$ coloring} if every hyperedge $e \in E$ is properly $(r,p)$ colored by the $r$-coloring. We study the maximum number of hyperedges that can be properly $(r,p)$ colored by a single $r$-coloring and the structures that maximizes number of properly $(r,p)$ colored hyperedges

Taylor Domination, Tur\'an lemma, and Poincar\'e-Perron Sequences

We consider "Taylor domination" property for an analytic function $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k},$ in the complex disk $D_R$, which is an inequality of the form $$|a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1.$$ This property is closely related to the classical notion of "valency" of $f$ in $D_R$. For $f$ - rational function we show that Taylor domination is essentially equivalent to a well-known and widely used Tur\'an's inequality on the sums of powers. Next we consider linear recurrence relations of the Poincar\'e type $$a_{k}=\sum_{j=1}^{d}[c_{j}+\psi_{j}(k)]a_{k-j},\ \ k=d,d+1,\dots,\quad\text{with }\lim_{k\rightarrow\infty}\psi_{j}(k)=0.$$ We show that the generating functions of their solutions possess Taylor domination with explicitly specified parameters. As the main example we consider moment generating functions, i.e. the Stieltjes transforms $$S_{g}\left(z\right)=\int\frac{g\left(x\right)dx}{1-zx}.$$ We show Taylor domination property for such $S_{g}$ when $g$ is a piecewise D-finite function, satisfying on each continuity segment a linear ODE with polynomial coefficients

Hypergraphs with vanishing Tur\'an density in uniformly dense hypergraphs

P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We study related questions for $3$-uniform hypergraphs with the additional assumption that $H$ has to be uniformly dense with respect to vertex sets. In particular, we characterise those hypergraphs $F$ that are guaranteed to appear in large uniformly dense hypergraphs $H$ of positive density. We also review the case when the density of the induced subhypergraphs of $H$ may depend on the proportion of the considered vertex sets.Comment: 26 pages, this version addresses changes arising from the referee report

Delsarte's Extremal Problem and Packing on Locally Compact Abelian Groups

Let G be a locally compact Abelian group, and let X, Y be two open sets in G. We investigate the extremal constant C(X,Y) defined to be the supremum of integrals of functions f from the class F(X,Y), where F(X,Y) is the family of positive definite functions f on G such that f(0) = 1, the positive part of f is supported in X, and its negative part is supported in Y. In the case when X=Y, the problem is exactly the so-called Tur\'an problem for the set X. When Y= G, i.e., there is a restriction only on the set of positivity of f, we obtain the Delsarte problem. The Delsarte problem is the sharpest Fourier analytic tool to study packing density by translates of a given "master copy" set, which was studied first in connection with packing densities of Euclidean balls. We give an upper estimate of the constant C(X,Y) in the situation when the set X satisfies a certain packing type condition. This estimate is given in terms of the asymptotic uniform upper density of sets in locally compact Abelian groups.Comment: The November 22 version is a throughout, but all in all slight revision of the original submissio

Hypergraph Removal Lemmas via Robust Sharp Threshold Theorems

The classical sharp threshold theorem of Friedgut and Kalai (1996) asserts that any symmetric monotone function $f:\{0,1\}^{n}\to\{0,1\}$ exhibits a sharp threshold phenomenon. This means that the expectation of $f$ with respect to the biased measure $\mu_{p}$ increases rapidly from 0 to 1 as $p$ increases. In this paper we present `robust' versions of the theorem, which assert that it holds also if the function is `almost' monotone, and admits a much weaker notion of symmetry. Unlike the original proof of the theorem which relies on hypercontractivity, our proof relies on a `regularity' lemma (of the class of Szemer\'edi's regularity lemma and its generalizations) and on the `invariance principle' of Mossel, O'Donnell, and Oleszkiewicz which allows (under certain conditions) replacing functions on the cube $\{0,1\}^{n}$ with functions on Gaussian random variables. The hypergraph removal lemma of Gowers (2007) and independently of Nagle, R\"odl, Schacht, and Skokan (2006) says that if a $k$-uniform hypergraph on $n$ vertices contains few copies of a fixed hypergraph $H$, then it can be made $H$-free by removing few of its edges. While this settles the `hypergraph removal problem' in the case where $k$ and $H$ are fixed, the result is meaningless when $k$ is large (e.g. $k>\log\log\log n$). Using our robust version of the Friedgut-Kalai Theorem, we obtain a hypergraph removal lemma that holds for $k$ up to linear in $n$ for a large class of hypergraphs. These contain all the hypergraphs such that both their number of edges and the sizes of the intersections of pairs of their edges are upper bounded by some constant.Comment: 46 page

Some remarks on the extremal function for uniformly two-path dense hypergraphs

We investigate extremal problems for hypergraphs satisfying the following density condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, \eta,P_2)$-dense if for any two subsets of pairs $P$, $Q\subseteq V\times V$ the number of pairs $((x,y),(x,z))\in P\times Q$ with $\{x,y,z\}\in E$ is at least $d|\mathcal{K}_{P_2}(P,Q)|-\eta|V|^3,$ where $\mathcal{K}_{P_2}(P,Q)$ denotes the set of pairs in $P\times Q$ of the form $((x,y),(x,z))$. For a given $3$-uniform hypergraph $F$ we are interested in the infimum $d\geq 0$ such that for sufficiently small $\eta$ every sufficiently large $(d, \eta,P_2)$-dense hypergraph $H$ contains a copy of $F$ and this infimum will be denoted by $\pi_{P_2}(F)$. We present a few results for the case when $F=K_k^{(3)}$ is a complete three uniform hypergraph on $k$ vertices. It will be shown that $\pi_{P_2}(K_{2^r}^{(3)})\leq \frac{r-2}{r-1}$, which is sharp for $r=2,3,4$, where the lower bound for $r=4$ is based on a result of Chung and Graham [Edge-colored complete graphs with precisely colored subgraphs, Combinatorica 3 (3-4), 315-324].Comment: 25 pages, dedicated to Ron Graham on the occasion of his 80th birthda

On hypergraphs without loose cycles

Recently, Mubayi and Wang showed that for $r\ge 4$ and $\ell \ge 3$, the number of $n$-vertex $r$-graphs that do not contain any loose cycle of length $\ell$ is at most $2^{O( n^{r-1} (\log n)^{(r-3)/(r-2)})}$. We improve this bound to $2^{O( n^{r-1} \log \log n) }$.Comment: 6 page

The Density Tur\'an problem

Let $H$ be a graph on $n$ vertices and let the blow-up graph $G[H]$ be defined as follows. We replace each vertex $v_i$ of $H$ by a cluster $A_i$ and connect some pairs of vertices of $A_i$ and $A_j$ if $(v_i,v_j)$ was an edge of the graph $H$. As usual, we define the edge density between $A_i$ and $A_j$ as $d(A_i,A_j)=\frac{e(A_i,A_j)}{|A_i||A_j|}.$ We study the following problem. Given densities $\gamma_{ij}$ for each edge $(i,j)\in E(H)$. Then one has to decide whether there exists a blow-up graph $G[H]$ with edge densities at least $\gamma_{ij}$ such that one cannot choose a vertex from each cluster so that the obtained graph is isomorphic to $H$, i.e, no $H$ appears as a transversal in $G[H]$. We call $d_{crit}(H)$ the maximal value for which there exists a blow-up graph $G[H]$ with edge densities $d(A_i,A_j)=d_{crit}(H)$ $((v_i,v_j)\in E(H))$ not containing $H$ in the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs

Extremal problems in uniformly dense hypergraphs

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem in extremal combinatorics. While for graphs ($k=2$) this problem is well understood, due to the work of Mantel, Tur\'an, Erd\H{o}s, Stone, Simonovits and many others, only very little is known for $k$-uniform hypergraphs for $k>2$. Already the case when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices is still wide open even for $k=3$. We consider variants of such problems where the large hypergraph $H$ enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H{o}s and S\'os about 30 years ago. In recent work with R\"odl and Schacht it turned out that the regularity method for hypergraphs, established by Gowers and by R\"odl et al. about a decade ago, is a suitable tool for extremal problems of this type and we shall discuss some of those recent results and some interesting open problems in this area.Comment: 30 pages; surve