In this note we describe how Lasoń's generalization of Alon's Combinatorial Nullstellensatz gives a framework for constructing lower bounds on the Turán number $\operatorname{ex}(n, K^{(r)}_{s_1,\dots,s_r})$ of the complete $r$-partite $r$-uniform hypergraph $K^{(r)}_{s_1,\dots,s_r}$. To illustrate the potential of this method, we give a short and simple explicit construction for the Erdős box problem, showing that $\operatorname{ex}(n, K^{(r)}_{2,\dots,2}) = Ω(n^{r - 1/r})$, which asymptotically matches best known bounds when $r \leq 4$.
The long-standing Erdős-Faber-Lovász conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős-Faber-Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations,…
Let $k\ge 2$ and $n_1\ge n_2\ge n_3\ge n_4$ be integers such that $n_4$ is sufficiently larger than $k$. We determine the maximum number of edges of a 4-partite graph with parts of sizes $n_1,\dots, n_4$ that does not contain $k$ vertex-disjoint triangles. For any $r> t\ge 3$, we give a conjecture on the maximum number of edges of an $r$-partite graph that does not contain $k$ vertex-disjoint cliques $K_t$.
For two graphs $G$ and $H$, the Turán number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Turán numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this paper we will determine the Turán numbers $ex(K_{a_1,\ldots,a_r},kK_r)$ for all $r\geq 3$ and $k\geq 1$.
We consider a problem proposed by Linial and Wilf to determine the structure of graphs that allows the maximum number of $q$-colorings among graphs with $n$ vertices and $m$ edges. Let $T_r(n)$ denote the Turán graph - the complete $r$-partite graph on $n$ vertices with partition sizes as equal as possible. We prove that for all odd integers $q\geq 5$ and sufficiently large $n$, the Turán graph $T_2(n)$ has at least as many $q$-colorings as any other graph $G$ with the same number of vertices and edges as $T_2(n)$, with equality holding if and only if $G=T_2(n)$. Our proof builds on methods by…
We enumerate labelled and unlabelled Hamiltonian cycles in complete $n$-partite graphs $K_{d,d,\ldots,d}$ having exactly $d$ vertices in each part (in other words, Turán graphs $T(nd, n))$. We obtain recurrence relations that allow us to find the exact values $b_{n}^{(d)}$ of such cycles for arbitrary $n$ and $d$
Turán's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability Theorem of Erdős and Simonovits shows that if a $K_{r+1}$-free graph with $n$ vertices has close to the maximal $t_r(n)$ edges, then it is close to being $r$-partite. In this paper we determine exactly the $K_{r+1}$-free graphs with at least $m$ edges that are farthest from being $r$-partite, for any $m\ge t_r(n) - δ_r n^2$. This extends work by Erdős, Győri and Simonovits, and proves a conjecture of Balogh, Clemen, Lavrov, Lidický and Pfender.
Kalai conjectured that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r} {n \choose r-1}$ edges contains all tight $r$-trees of some fixed size $t$. We prove Kalai's conjecture for $r$-partite $r$-uniform hypergraphs. Our result is asymptotically best possible up to replacing the term $\frac{t-1}{r}$ with the term $\frac{t-r+1}{r}$. We apply our main result in graphs to show an upper bound for the Turán number of trees.
The exact Turán number of the even wheel $W_{2k+2}$ among non-$3$-partite graphs
School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）
For two $s$-uniform hypergraphs $H$ and $F$, the Turán number $ex_s(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Let $s, r, k, n_1, \ldots, n_r$ be integers satisfying $2\leq s\leq r$ and $n_1\leq n_2\leq \cdots\leq n_r$. De Silva, Heysse and Young determined $ex_2(K_{n_1, \ldots, n_r}, kK_2)$ and De Silva, Heysse, Kapilow, Schenfisch and Young determined $ex_2(K_{n_1, \ldots, n_r},kK_r)$. In this paper, as a generalization of these results, we consider three Turán-type problems for $k$ disjoint cliques in $r$-partite $s$-uniform hypergraphs. First, we consider a multi-…
In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we prove a version of this result where we assume that $r$-tuples of vertices in $G$ have their neighbours evenly distributed in $G$. Third, we also prove a counting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$ satisfies our density bound, and present some open problems. A st…
Alspach [{\sl Bull. Inst. Combin. Appl.}~{\bf 52} (2008), 7--20] defined the maximal matching sequencibility of a graph $G$, denoted~$ms(G)$, to be the largest integer $s$ for which there is an ordering of the edges of $G$ such that every $s$ consecutive edges form a matching. In this paper, we consider the natural analogue for hypergraphs of this and related results and determine $ms(λ\mathcal{K}_{n_1,\ldots, n_k})$ where $λ\mathcal{K}_{n_1,\ldots, n_k}$ denotes the multi-$k$-partite $k$-graph with edge multiplicity $λ$ and parts of sizes $n_1,\ldots,n_k$, respectively. It turns out that thes…
We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{\text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Turán number $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$ $\bigg($ respectively, $\text{ex}(K_{n_{1},n_{2},n_{3}},\{C_{3}, C_{4}^{\text{multi}}\})$ $\bigg)$ is the maximum number of edges in a graph $G\subseteq K_{n_{1},n_{2},n_{3}}$ such that $G$ contains no $C_{4}^{\text{multi}}$ $\bigg($ respectively, $G$ contains neither $C_{3}$ nor $C_{4}^{\text{multi}}$ $\bigg)$. We call a $C^{multi}_4$ rainbow if a…
A zero-one matrix $M$ contains a zero-one matrix $A$ if one can delete some rows and columns of $M$, and turn some 1-entries into 0-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted by $ex(n,A)$, is the maximum number of $1$-entries in an $n\times n$ sized matrix $M$ that does not contain $A$. A matrix $A$ is column-$t$-partite (or row-$t$-partite), if it can be cut along the columns (or rows) into $t$ submatrices such that every row (or column) of these submatrices contains at most one $1$-entry. We prove that if $A$ is column-$t$-partite, then $ex(n,A)<n^{2-\…
We look at structures that must be removed (or reversed) in order to make acyclic a given oriented graph. For a directed acyclic graph $H$ and an oriented graph $G$, let $f_H(G)$ be the maximum number of pairwise disjoint copies of $H$ that can be found in {\em all} feedback arc sets of $G$. In particular, to make $G$ acyclic, one must remove (or reverse) $f_H(G)$ pairwise disjoint copies of $H$. Most intriguing is the case where $H$ is a $k$-clique, where the parameter is denoted by $f_k(G)$. Determining $f_k(G)$ for arbitrary $G$ seems challenging. Here we determine $f_k(G)$ precisely for al…
An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs $\mathcal{H}$ and $\mathcal{G}$, the anti-Ramsey number $ar(\mathcal{G}, \mathcal{H})$ of $\mathcal{H}$ in $\mathcal{G}$ is the maximum number of colors needed to color the edges of $\mathcal{G}$ so that there does not exist a rainbow copy of $\mathcal{H}$. Li et al. determined the anti-Ramsey number of $k$-matchings in complete bipartite graphs. Jin and Zang showed the uniqueness of the extremal coloring. In this paper, as a generalization of these results, we determine the anti-Ramsey numbe…
Given two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, a perfect $F$-tiling (or an $F$-factor) in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. For all complete $k$-partite $k$-graphs $K$, Mycroft proved a minimum codegree condition that guarantees a $K$-factor in an $n$-vertex $k$-graph, which is tight up to an error term $o(n)$. In this paper we improve the error term in Mycroft's result to a sub-linear term that relates to the Turán number of $K$ when the differences of the sizes of the vertex classes of $K$ are co-prime. Furthermore, we find …
Turan's Theorem states that every graph of a certain edge density contains a complete graph $K^k$ and describes the unique extremal graphs. We give a similar Theorem for l-partite graphs. For large l, we find the minimal edge density $d^k_l$, such that every $\ell$-partite graph whose parts have pairwise edge density greater than $d^k_l$ contains a $K^k$. It turns out that $d^k_l=(k-2)/(k-1)$ for large enough l. We also describe the structure of the extremal graphs. For the case of triangles we show that $d^3_{13}=1/2$, disproving a conjecture by Bondy, Shen, Thomasse and Thomassen.
A well-known special case of a conjecture attributed to Ryser states that k-partite intersecting hypergraphs have transversals of at most k-1 vertices. An equivalent form was formulated by Gyárfás: if the edges of a complete graph K are colored with k colors then the vertex set of K can be covered by at most k-1 sets, each connected in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: Z. Király proved that in every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >= 3), the vertex set of K^r can be covered by at most $\lceil k/r \…
The generalized Turán number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $χ(H)<χ(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the extremal graph is a complete $(χ(F)-1)$-partite graph. We obtain multiple exact results with other kinds of extremal graphs.
Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Turán numbers of $k$-dimensional simplicial complexes: for each $k \in \mathbb N$, there is a $λ_k \ge k^{-2k^2}$ such that for any $k$-complex $\mathcal{S}$, every $k$-complex on $n \ge n_0(\mathcal{S})$ vertices with at least $n^{k+1 - λ_k}$ facets contains a homeomorphic copy of $\mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $λ_1$ is a result of Mader from 1967, and the existence of $λ_2$ was suggested by L…
Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.
For graphs $H$ and $F$, the generalized Turán number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We say that $H$ is $F$-Turán-good if $ex(n,H,F)$ is the number of copies in the $(χ(F)-1)$-partite Turán graph, provided $n$ is large enough. We present a general theorem in case $F$ has an edge whose deletion decreases the chromatic number. In particular, this determines $ex(n,P_k,C_{2\ell+1})$ and $ex(n,C_{2k},C_{2\ell+1})$ exactly, if $n$ is large enough. We also study the case when $F$ has a vertex whose deletion decreases the chromatic number.
A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to have a monochromatic solution whenever the positive integers are finitely colored. In this paper, we provide a variant of this theorem. For $k \ge 2$, we present conditions such that, when the set of variables is partitioned into $k$ subsets, there is a solution such that the variables of each subset are monochromatic, which we call a semi-monochromatic sol…
The main contribution of this paper is a formula for the number of acyclic orientations of a complete bipartite, $K_{n_1,n_2},$ revealing that it is equal to the poly-Bernoulli number $B_{n_1}^{(-n_2)}$ introduced in 1997 by Kaneko. We also give a simple bijective identification of acyclic orientations and lonesum matrices, confirming a 2008 result of Brewbaker, and show that the poly-Bernoulli numbers behave concavely, as well being symmetric. A second goal is to explore the behaviour of more general complete $r$-partite graphs in the space of the number of acyclic orientations. We prove that…
Alon, Balogh, Keevash and Sudakov proved that the $(k-1)$-partite Turán graph maximizes the number of distinct $r$-edge-colorings with no monochromatic $K_k$ for all fixed $k$ and $r=2,3$, among all $n$-vertex graphs. In this paper, we determine this function asymptotically for $r=2$ among $n$-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Turán theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with su…
Stability from graph symmetrization arguments in generalized Turán problems
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Department of Computer Science and Information Theory, Faculty of Electrical Engineering and Informatics, Budapest University of Technology and Economics（布达佩斯科技大学电气与信息工程学院计算机科学与信息论系）
Combinatorial batch codes provide a tool for distributed data storage, with the feature of keeping privacy during information retrieval. Recently, Balachandran and Bhattacharya observed that the problem of constructing such uniform codes in an economic way can be formulated as a Turán-type question on hypergraphs. Here we establish general lower and upper bounds for this extremal problem, and also for its generalization where the forbidden family consists of those $r$-uniform hypergraphs $H$ which satisfy the condition $k\ge |E(H)|> |V(H)|+q$ (for $k>q+r$ and $q> -r$ fixed). We also prove that…
A Berge-$K_4$ in a triple system is a configuration with four vertices $v_1,v_2,v_3,v_4$ and six distinct triples $\{e_{ij}: 1\le i< j \le 4\}$ such that $\{v_i,v_j\}\subset e_{ij}$ for every $1\le i<j\le 4$. We denote by $\cal{B}$ the set of Berge-$K_4$ configurations. A triple system is $\cal{B}$-free if it does not contain any member of $\cal{B}$. We prove that the maximum number of triples in a $\cal{B}$-free triple system on $n\ge 6$ points is obtained by the balanced complete $3$-partite triple system: all triples $\{abc: a\in A, b\in B, c\in C\}$ where $A,B,C$ is a partition of $n$ poin…
An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Turán graph, which is the complete $r$-partite graph on $n$ vertices with part sizes that differ by at most one. The well-known Turán Theorem states that $T_{n,r}$ is the only extremal graph for complete graph $K_{r+1}$. Erdös et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we dete…
A d-partite hypergraph is called *fractionally balanced* if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall's theorem we prove lower bounds on the matching number of such hypergraphs. These bounds yield rainbow versions of the KKM theorem for products of simplices, which in turn are used to obtain some results on multiple-cake division, and on rainbow matchings in families of d-intervals.
For a 2-graph $F$, let $H_F^{(r)}$ be the $r$-graph obtained from $F$ by enlarging each edge with a new set of $r-2$ vertices. We show that if $χ(F)=\ell>r \geq 2$, then $ {\rm ex}(n,H_F^{(r)})= t_r (n,\ell-1)+ Θ( {\rm biex}(n,F)n^{r-2}),$ where $t_r (n,\ell-1)$ is the number of edges of an $n$-vertex complete balanced $\ell-1$ partite $r$-graph and ${\rm biex}(n,F)$ is the extremal number of the decomposition family of $F$. Since ${\rm biex}(n,F)=O(n^{2-γ})$ for some $γ>0$, this improves on the bound ${\rm ex}(n,H_F^{(r)})= t_r (n,\ell-1)+ o(n^r)$ by Mubayi (2016). Furthermore, our result imp…
Recently, several hypergraph Turán problems were solved by the powerful random algebraic method. However, the random algebraic method usually requires some parameters to be very large, hence we are concerned about how these Turán numbers depend on such large parameters of the forbidden hypergraphs. In this paper, we determine the dependence on such specified large constant for several hypergraph Turán problems. More specifically, for complete $r$-partite $r$-uniform hypergraphs, we show that if $s_{r}$ is sufficiently larger than $s_{1},s_{2},\ldots,s_{r-1},$ then $$\textup{ex}_{r}(n,K_{s_{1},…
For two graphs $F$ and $H$, the relative Turán number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these quantities as a function of the maximum degree of $H$. In this paper, we study a generalization for uniform hypergraphs. If $F$ is a complete $r$-partite $r$-uniform hypergraph with parts of sizes $s_1,s_2,\dots,s_r$ with each $s_{i + 1}$ sufficiently large relative to $s_i$, then with $1/β= \sum_{i = 2}^r \prod_{j = 1}^{i - 1} s_j$ we prove that for any $r$-uniform hyp…
For $s<r$, let $B_{r,s}$ be the graph consisting of two copies of $K_r$, which share exactly $s$ vertices. Denote by $ex(n, K_r, B_{r,s})$ the maximum number of copies of $K_r$ in a $B_{r,s}$-free graph on $n$ vertices. In 1976, Erdős and Sós determined $ex(n,K_3,B_{3,1})$. Recently, Gowers and Janzer showed that $ex(n,K_r,B_{r,r-1})=n^{r-1-o(1)}$. It is a natural question to ask for $ex(n,K_r,B_{r,s})$ for general $r$ and $s$. In this paper, we mainly consider the problem for $s=1$. Utilizing the Zykov's symmetrization, we show that $ex(n,K_4, B_{4,1})=\lfloor (n-2)^2/4\rfloor$ for $n\geq 45$…
Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. For such a graph $F,$ a classical result of Simonovits from 1966 shows that every graph on $n\ge n_0(F)$ vertices with more than $\frac{χ(F)-2}{χ(F)-1}\cdot \frac{n^2}{2}$ edges contains a copy of $F$. In this paper we derive a similar theorem for multipartite graphs. For a graph $H$ and an integer $\ell \geq v(H)$, let $d_{\ell}(H)$ be the minimum real number such that every $\ell$-partite graph whose edge density between any two parts is greater than $d_{\ell}(H)$ contains a copy of $H$. Our main contribut…
Given a family of $r$-uniform hypergraphs ${\cal F}$ (or $r$-graphs for brevity), the Turán number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain any member of ${\cal F}$. A pair $\{u,v\}$ is covered in a hypergraph $G$ if some edge of $G$ contains $\{u,v\}$. Given an $r$-graph $F$ and a positive integer $p\geq n(F)$, let $H^F_p$ denote the $r$-graph obtained as follows. Label the vertices of $F$ as $v_1,\ldots, v_{n(F)}$. Add new vertices $v_{n(F)+1},\ldots, v_p$. For each pair of vertices $v_i,v_j$ not covered in $F$, add a…
Spectral extremal results for triangle-free graphs with chromatic number at least four
School of Mathematics, East China University of Science and Technology（华东理工大学数学学院）；School of Mathematics and Physics, Xinjiang Institute of Engineering（新疆工程学院数理学院）
For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of $\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. The determination of Turán number $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ has become the fundamental core problem in extremal graph theory ever since the pioneering work Turán's Theorem was published in $1941$. Although we have some rich results for the simple graph case, only sporadic results have been known for the hypergraph Turán problems. In this paper, we mainly focus on the function $\text{ex…
