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 contribution is to show that $d_{\ell}(H)=\frac{χ(H)-2}{χ(H)-1}$ for $\ell \ge \ell_0(H)$ sufficiently large if and only if $H$ admits a vertex-colouring with $χ(H)-1$ colours such that all colour classes but one are independent sets, and the exceptional class induces just a matching. When $H$ is a clique, this recovers a result of Pfender [Complete subgraphs in multipartite graphs, Combinatorica 32 (2012), 483--495]. We also consider several extensions of Pfender's result.
Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove corresponding stability theorems. These results complete work started by Erdos in 1963.
The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine the Turán number of the extension of an $r$-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Turán number for $r=3$. As the key ingredient of the proof we show that the Lagrangian of intersecting $r$-graphs is maximized by principally intersecting $r$-graphs for $r \geq 4$.
Localization of spectral Turán theorems for signed graphs
School of Mathematics and Statistics,Northwestern Polytechnical University（西北工业大学数学与统计学院）；Xi’an-Budapest Joint Research Center for Combinatorics,Northwestern Polytechnical University（西北工业大学西安-布达佩斯组合学联合研究中心）
We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results.
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.
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.
In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Turán's theorem using the connection. The connection of Lagrangians and Turán densities can be also used to prove the fundamental theorem of Erdös-Stone-Simonovits on Turán densities of graphs. Very recently, the study of Turán densities of non-uniform hypergraphs have been motivated by extremal poset problems. In this paper, we attempt to explore the applications of Lagrangian method in determining Turán densities of non-uniform hyperg…
On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Turán Theorem
School of Mathematics, Shandong University（山东大学数学学院）；Extremal Combinatorics and Probability Group, Institute for Basic Science（基础科学研究院极值组合与概率组）；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）
In 1952, Dirac proved that every $2$-connected $n$-vertex graph with the minimum degree $k+1$ contains a cycle of length at least $\min\{n, 2(k+1)\}$. Here we obtain a stability version of this result by characterizing those graphs with minimum degree $k$ and circumference at most $2k+1$. We present applications of the above-stated result by obtaining generalized Turán numbers. In particular, for all $\ell \geq 5$ we determine how many copies of a five-cycle as well as four-cycle are necessary to guarantee that the graph has circumference larger than $\ell$. In addition, we give a new proof of…
Let $\mc{K}_{3,3}^3$ be the 3-graph with 15 vertices $\{x_i, y_i: 1 \le i \le 3\}$ and $\{z_{ij}: 1 \le i,j \le 3\}$, and 11 edges $\{x_1, x_2, x_3\}$, $\{y_1, y_2, y_3\}$ and $\{\{x_i, y_j, z_{ij}\}: 1 \le i,j \le 3\}$. We show that for large $n$, the unique largest $\mc{K}_{3,3}^3$-free 3-graph on $n$ vertices is a balanced blow-up of the complete 3-graph on 5 vertices. Our proof uses the stability method and a result on lagrangians of intersecting families that has independent interest.
We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for $t \ge 3$, we show that any two-colouring of the complete graph on $n$ vertices that is $δ$-far from being monochromatic contains an \emph{unavoidable $t$-colouring} when $δ\gg n^{-1/t}$, where an unavoidable $t$-colouring is any two-colouring of a clique of order $2t$ in which one colour forms either a clique of order $t$ or two disjoint cliques of order $t$. Next, for $ t\ge 3$, we show that any tournament on $n$ vertices that is $δ$-far from being transitive contains an \emph{u…
Confirming a conjecture of Vera T. Sós in a very strong sense, we give a complete solution to Turán's hypergraph problem for the Fano plane. That is we prove for $n\ge 8$ that among all $3$-uniform hypergraphs on $n$ vertices not containing the Fano plane there is indeed exactly one whose number of edges is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for $n=7$ there is exactly one other extremal configuration with the same number of edges: the hypergraph arising from a clique of order $7$ by removing all five edges containing a fixed pair of vertices. For sufficient…
Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turan-type results) or on finding spanning subgraphs (Dirac-type results). In this paper we are interested in finding intermediate-sized subgraphs. We investigate minimum degree conditions under which a graph G contains squared paths and squared cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends results of Fan and Kierstead [J. Combin. Theory Ser. B 63 (1995), 55--64] and of Komlos, Sarkozy, and Szemeredi [Random Struct…
Turán-type extremal problems for unbalanced signed graphs
Northwestern Polytechnical University（西北工业大学）；Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University（西安-布达佩斯组合学联合研究中心）；School of Mathematical Sciences & Academy for Multidisciplinary Studies, Captial Normal University（首都师范大学数学科学学院与跨学科研究院）
Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), 77--97] showed that the uniform Turán density of every $3$-uniform hypergraph is either $0$ or at least $1/27$, and asked whether there exist $3$-uniform hypergraphs with uniform Turán density equal or arbitrarily close to $1/27$. We construct $3$-uniform hypergraphs with uniform Turán density equal to $1/27$.
Given $r$-uniform hypergraphs $G$ and $H$ the Turán number $\rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erdős-Rényi random $r$-uniform hypergraph, and $H=C_{2\ell}^{(r)}$, the $r$-uniform linear cycle of length $2\ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine $\rm ex(G_{n,p}^{(r)}, C_{2\ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $\ell$ …
A signed graph $Σ= (G, σ)$ is a graph where the function $σ$ assigns either $1$ or $-1$ to each edge of the simple graph $G$. The adjacency matrix of $Σ$, denoted by $A(Σ)$, is defined canonically. In a recent paper, Wang et al. extended the eigenvalue bounds of Hoffman and Cvetković for the signed graphs. They proposed an open problem related to the balanced clique number and the largest eigenvalue of a signed graph. We solve a strengthened version of this open problem. As a byproduct, we give alternate proofs for some of the known classical bounds for the least eigenvalues of the unsigned gr…
Vertex-colored Turán theorems with applications in extremal hypergraph problems
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）；School of Data Science and Engineering, South China Normal University（华南师范大学数据科学与工程学院）；Center for Discrete Mathematics, Fuzhou University（福州大学离散数学研究中心）
In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least $d$ contains $H$. In particular, they raise the questions of determining the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. The former question was solved only recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter sti…
We give a purely combinatorial proof of the density Hales--Jewett Theorem that is modeled after Polymath's proof but is significantly simpler. In particular, we avoid the use of the equal-slices measure and work exclusively with the uniform measure.
This is an exposition of the combinatorial proof of the density Hales--Jewett theorem, due to D.\,H.\,J. Polymath in 2012. The theorem says that for given $\de>0$ and $k$, for every $n>n_0$ every set $A\sus\{1,2,\ds,k\}^n$ with $|A|\ge\de k^n$ contains a combinatorial line. It implies Szemerédi's theorem, which claims that for given $\de>0$ and $k$, for every $n>n_0$ every set $A\sus\{1,2,\ds,n\}$ with $|A|\ge\de n$ contains a $k$-term arithmetic progression.
Let $\mathcal{H}$ be an $r$-uniform hypergraph. The Turán number $\text{ex}(n,\mathcal{H})$ is the maximum number of edges in an $n$-vertex $\mathcal{H}$-free $r$-uniform hypergraph. The Turán density of $\mathcal{H}$ is defined by \[π(\mathcal{H})=\lim_{n\rightarrow\infty}\frac{\text{ex}(n,\mathcal{H})}{\binom{n}{r}}.\] In this paper, we consider the Turán density of projective geometries. We give two new constructions of $PG_{m}(q)$-free hypergraphs which improve some results given by Keevash (J. Combin. Theory Ser. A, 111: 289--309, 2005). Based on an upper bound of blocking sets of $PG_m(q…
We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi-Hajnal Theorem.
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic progressions.
For every positive integer $k$, we show that every graph of order $n$ at least $3k$ with more than $$\max\{{2k-1\choose 2}+(2k-1)(n-(2k-1)),{3k-1\choose 2}+(n-(3k-1))\}$$ edges has $k$ vertex disjoint cycles, which is a best possible density version of a theorem of Corrádi and Hajnal.
In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements.
Let $G$ be a connected graph of order $n$ with vertex set $V(G)$. A subset $S\subseteq V(G)$ is an $(a,b)$-dominating set if every vertex $v\in S$ is adjacent to at least $a$ vertices in $S$ and every $v\in V\setminus S$ is adjacent to at least $b$ vertices in $S$. The minimum cardinality of an $(a,b)$-dominating set of $G$ is the $(a,b)$-domination number of $G$, denoted by $γ_{a,b}(G)$. There are various results about upper bounds for $γ_{a,b}(G)$ when $G$ is regular or $a$ and $b$ are small numbers. In the first part of this paper, for a given graph $G$ with the minimum degree of $\max\{a,b…
For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \geq b_r(G)$ for any $G$, and Turán's Theorem says that equality holds for complete graphs. With $G_{n,p}$ the usual ("binomial" or "Erdős-Rényi") random graph, we show: For each fixed r there is a C such that if \[ p=p(n) > Cn^{-\tfrac{2}{r+1}}\log^{\tfrac{2}{(r+1)(r-2)}}n, \] then $\Pr(t_r(G_{n,p})=b_r(G_{n,p}))\rightarrow 1$ as $n\rightarrow\infty$. This is best possible (apart from the value of $C$) and settles a question first considered by Babai…
Given a measurable set $A\subset \mathbb R^d$ we consider the "large-distance graph" $\mathcal{G}_A$, on the ground set $A$, in which each pair of points from $A$ whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the $2d$-dimensional Lebesgue measure of the edge set as well as the $d$-dimensional Lebesgue measure of the vertex set of a large-distance graph in the $d$-dimensional Euclidean space that contains no copies of a complete graph on $k$ vertices. The former problem may be seen as a continuous analogue of Turán's classical graph theorem, and the latte…
We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.
We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán's theorem, Szemerédi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Turán's theorem to the random setting by showing that for every $ε> 0$ and every positive integer $t \geq 3$ there exists a constant $C$ such that, if $G$ is a random graph on $n$ vertices where each edge is chosen independently with probability at least $C n^{-2/(t+1)}$, then, with probability tending to $1$ as $n$ tends to infinity, every subgra…
We present a new proof of the Kővári-Sós-Turán theorem that $ex(n, K_{s,t}) = O(n^{2-1/t})$ for $s, t \geq 2$. The new proof is elementary, avoiding the use of convexity. For any $d$-uniform hypergraph $H$, let $ex_d(n,H)$ be the maximum possible number of edges in an $H$-free $d$-uniform hypergraph on $n$ vertices. Let $K_{H, t}$ be the $(d+1)$-uniform hypergraph obtained from $H$ by adding $t$ new vertices $v_1, \dots, v_t$ and replacing every edge $e$ in $E(H)$ with $t$ edges $e \cup \left\{v_1\right\},\dots, e \cup \left\{v_t\right\}$ in $E(K_{H, t})$. If $H$ is the $1$-uniform hypergraph …
