In 1981, Erdős and Simonovits conjectured that for any bipartite graph $H$ we have $\mathrm{ex}(n,H)=O(n^{3/2})$ if and only if $H$ is $2$-degenerate. Later, Erdős offered 250 dollars for a proof and 500 dollars for a counterexample. In this paper, we disprove the conjecture by finding, for any $\varepsilon>0$, a $3$-regular bipartite graph $H$ with $\mathrm{ex}(n,H)=O(n^{4/3+\varepsilon})$.
The cube Q is the usual 8-vertex graph with 12 edges. Here we give a new proof for a theorem of Erdős and Simonovits concerning the Turán number of the cube. Namely, it is shown that e(G) < n^{8/5}+(2n)^{3/2} holds for any n-vertex cube-free graph G. Our aim is to give a self-contained exposition. We also point out the best known results and supply bipartite versions.
The following sharpening of Turán's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there exists an (at most) $p$-chromatic subgraph $H_0$ such that $e(H_0)\geq e(G)-t$. Using this result we present a concise, contemporary proof (i.e., one applying Szemerédi's regularity lemma) for the classical stability result of Simonovits.
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.
The classical stability theorem of Erdős and Simonovits states that, for any fixed graph with chromatic number $k+1 \ge 3$, the following holds: every $n$-vertex graph that is $H$-free and has within $o(n^2)$ of the maximal possible number of edges can be made into the $k$-partite Turán graph by adding and deleting $o(n^2)$ edges. In this paper, we prove sharper quantitative results for graphs $H$ with a critical edge, both for the Erdős-Simonovits Theorem (distance to the Turán graph) and for the closely related question of how close an $H$-free graph is to being $k$-partite. In many cases, t…
An edge-spectral Erdős-Stone-Simonovits theorem and its stability
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（极值组合与概率研究组（ECOPRO），基础科学研究院（IBS））；Department of Mathematics, Stanford University（斯坦福大学数学系）
本文证明无 F 图的谱半径平方与边数之间的统一上界，建立相应的边谱稳定性，并改进关于大谱半径图中公共邻点数量的最优下界。
For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erdős and Simonovits \cite{ESi1} conjectured that for every family $\mathcal{F}$ of bipartite graphs, there exists $k$ such that $\ex{n}{\mathcal{F} \cup \mathcal{C}_k} \sim \ex{n}{\mathcal{F} \cup \mathcal{C}}$ as $n \rightarrow \infty$. This conjecture was proved by Erdős and Simonovits when $\mathcal{F} = \{C_4\}$, and for certain families of even cycles in \cite{KSV}. In this paper, we give a general approach …
A classical result of Bondy and Simonovits in extremal graph theory states that if a graph on $n$ vertices contains no cycle of length $2k$ then it has at most $O(n^{1+1/k})$ edges. However, matching lower bounds are only known for $k=2,3,5$. In this paper we study ordered variants of this problem and prove some tight estimates for a certain class of ordered cycles that we call bordered cycles. In particular, we show that the maximum number of edges in an ordered graph avoiding bordered cycles of length at most $2k$ is $Θ(n^{1+1/k})$. Strengthening the result of Bondy and Simonovits in the cas…
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 consider this problem when both $H$ and $F$ have at most four vertices. We give sharp results in almost all cases, and connect the remaining cases to well-known unsolved problems. Our main new contribution is applying the progressive induction method of Simonovits for generalized Turán problems.
Let $\mc{F}$ be a family of graphs. A graph is {\em $\mc{F}$-free} if it contains no copy of a graph in $\mc{F}$ as a subgraph. A cornerstone of extremal graph theory is the study of the {\em Turán number} $ex(n,\mc{F})$, the maximum number of edges in an $\mc{F}$-free graph on $n$ vertices. Define the {\em Zarankiewicz number} $z(n,\mc{F})$ to be the maximum number of edges in an $\mc{F}$-free {\em bipartite} graph on $n$ vertices. Let $C_k$ denote a cycle of length $k$, and let $\mc{C}_k$ denote the set of cycles $C_{\ell}$, where $3 \le \ell \leq k$ and $\ell$ and $k$ have the same parity. …
We study the behaviour of $K_{r+1}$-free graphs $G$ of almost extremal size, that is, typically, $e(G)=ex(n,K_{r+1})-O(n)$. We show that such graphs must have a large amount of 'symmetry', in particular that all but very few vertices of $G$ must have twins. As a corollary, we obtain a new, short proof of a theorem of Simonovits on the structure of extremal graphs with $ω(G)\leq r$ and $χ(G)\geq k$ for fixed $k \geq r \geq 2$.
The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) = Θ(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of…
Given a graph $H$ and a positive integer $n$, the {\it Turán number} $\ex(n,H)$ is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. A real number $r\in(1,2)$ is called a {\it Turán exponent} if there exists a bipartite graph $H$ such that $\ex(n,H)=Θ(n^r)$. A long-standing conjecture of Erdős and Simonovits states that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q> p$. In this paper, we build on recent developments on the conjecture to establish a large family of new Turán exponents. In particular, it follows from o…
Both the Simonovits stability theorem and the Nikiforov spectral stability theorem are powerful tools for solving exact values of Turán numbers in extremal graph theory. Recently, Füredi [J. Combin. Theory Ser. B 115 (2015)] provided a concise and contemporary proof of the Simonovits stability theorem. In this note, we present a unified treatment for some extremal graph problems, including short proofs of Nikiforov's spectral stability theorem and the clique stability theorem proved recently by Ma and Qiu [European J. Combin. 84 (2020)]. Moreover, some spectral extremal problems related to the…
The Turán number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let P_k be the path with k vertices, the square P^2_k of P_k is obtained by joining the pairs of vertices with distance one or two in P_k. The powerful theorem of Erdős, Stone and Simonovits determines the asymptotic behavior of ex(n,P^2_k). In the present paper, we determine the exact value of ex(n,P^2_5) and ex(n,P^2_6) and pose a conjecture for the exact value of ex(n,P^2_k).
Given a graph $H$ and a positive integer $n$, the Turán number of $H$ for the order $n$, denoted $ex(n,H)$, is the maximum size of a simple graph of order $n$ not containing $H$ as a subgraph. Given graphs $G$ and $H$, the notation $G \vee H$ means the joint of $G$ and $H$. $χ(G)$ denotes the chromatic number of a graph $G$. Since $χ(K_m \vee C_{2k-1})=m+3$ and there is an edge $e\in E(K_m \vee C_{2k-1})$ such that $χ(K_m \vee C_{2k-1}-e)= m+2$, by the Simonovits theorem, $ex(n, K_m \vee C_{2k-1}) = \lfloor \frac{(m+1)n^2}{2(m+2)}\rfloor$ for sufficiently large $n$. In this paper, we prove tha…
Given an $(r+1)$-chromatic graph $F$ and a graph $H$ that does not contain $F$ as a subgraph, we say that $H$ is strictly $F$-Turán-good if the Turán graph $T_{r}(n)$ is the unique graph containing the maximum number of copies of $H$ among all $F$-free graphs on $n$ vertices for every $n$ large enough. Győri, Pach and Simonovits (1991) proved that cycle $C_4$ of length four is strictly $K_{r+1}$-Turán-good for all $r\geq 2$. In this article, we extend this result and show that $C_4$ is strictly $F$-Turán-good, where $F$ is an $(r+1)$-chromatic graph with $r\ge 2$ and a color-critical edge. Mor…
Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G where f(n) is larger than the maximum number of vertices in a $K_t$-free induced subgraph of G. Erdős, Hajnal, Simonovits, Sós, and Szemerédi posed several open questions about RT_t(n,K_s,o(n)), among them finding the minimum s such that $RT_t(n,K_{t+s},o(n)) = Ω(n^2)$, where it is easy to see that $RT_t(n,K_{t+1},o(n)) = o(n^2)$. In this paper, we answer this question by proving that $RT_t(n,K_{t+2},o(n)) = Ω(n^2)$; our …
For a graph $H$, the Turán number of $H$, denoted by ex$(n,H)$, is the maximum number of edges of an $n$-vertex $H$-free graph. Let $g(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of $H$ in a $2$-edge-coloring of $K_n$. A wheel $W_m$ is a graph formed by connecting a single vertex to all vertices of a cycle of length $m-1$. The Turán number of $W_{2k}$ was determined by Simonovits in the 1960s. In this paper, we determine ex$(n,W_{2k+1})$ when $n$ is sufficiently large. We also show that, for sufficiently large $n$, $g(n,W_{2k+1})=\mbox{ex}(n,W_{2k+1})$ whic…
The Turán number of a graph $H$, $ex(n,H)$, is the maximum number of edges in any graph of order $n$ which does not contain $H$ as a subgraph. Lidický, Liu and Palmer determined $ex(n, F_m)$ for $n$ sufficiently large and proved that the extremal graph is unique, where $F_m$ is disjoint paths of $P_{k_1}, \ldots, P_{k_m}$ [Lidický,B., Liu,H. and Palmer,C. (2013). On the Turán number of forests. Electron. J. Combin. 20(2) Paper 62, 13 pp]. In this paper, by mean of a different approach, we determine $ex(n, F_m)$ for all integers $n$ with minor conditions, which extends their partial results. Fu…
In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the edge-order. A subgraph of an edge-ordered graph is itself an edge-ordered graph with the induced edge-order. We say that an edge-ordered graph $G$ $\textit{avoids}$ another edge-ordered graph $H$, if no subgraph of $G$ is isomorphic to $H$. The $\textit{Turán number}$ of an edge-ordered graph $H$ is the maximum number of edges in an edge-ordered graph on $n$ ver…
We describe the C_{2k+1}-free graphs on n vertices with maximum number of edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1. The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall, and Bollobas, but here we give a new streamlined proof. The complete determination of the extremal graphs is also new. We obtain that the bound for n_0(C_{2k+1}) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turan graph.
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…
Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Turán number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $α_s(G) < f$, where $α_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Turán number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $RT_s(n,K_t, n^δ)$ for fixed $δ<1$. We show that for an arbitrarily small $\varepsilon>0$ and $1/2<δ< 1$, $RT_s(n,K_{s+1}, n^δ) = Ω(n^{1+δ-\varepsilon})$…
Given graphs $H_1,\ldots, H_k$, a graph $G$ is $(H_1,\ldots, H_k)$-free if there is a $k$-edge-colouring $ϕ:E(G)\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function $f(n)$, the Ramsey-Turán function $\textrm{RT}(n,H_1,\ldots,H_k,f(n))$ is the maximum number of edges in an $n$-vertex $(H_1,\ldots,H_k)$-free graph with independence number at most $f(n)$. We determine $\textrm{RT}(n,K_3,K_s,δn)$ for $s\in\{3,4,5\}$ and sufficiently small $δ$, confirming a conjecture of Erdős and Sós from 1979. It is known that $\textrm{RT}(n,K_8,f(n))$ h…
The {\em Turán number} of an $r$-uniform graph $F$, denoted by $ex(n,F)$, is the maximum number of edges in an $F$-free $r$-uniform graph on $n$ vertices. The {\em Turán density} of $F$ is defined as $π(F)=\underset{n\rightarrow\infty}{\lim}{ex(n,F) \over {n \choose r }}.$ For graphs, Erdős-Stone-Simonovits (\cite{ESi}, \cite{ES}) showed that $Π_{\infty}^{(2)}=Π_{fin}^{(2)}=Π_{1}^{(2)}=\{0, {1 \over 2}, {2 \over 3}, \ldots,{l-1 \over l}, ...\}.$ We know quite few about the Turán density of an $r$-uniform graph for $r\ge 3$. Baber and Talbot \cite{BT}, and Pikhurko \cite{Pikhurko2} showed that …
What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from~$1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we gi…
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…
The problem of determining the Turán number of $C_4$ is a well studied problem that dates back to a paper of Erdös from 1938. It is known that Sidon sets can be used to construct $C_4$-free graphs. If $\A$ is a Sidon set in the abelian group $X$, the sum graph $G_{X, \A}$ with vertex set $X$ and edges set $E=\{\{x, y\}:x\neq y, x+y\in \A\}$ is $C_4$-free. Using the sum graph of a Sidon set of type Singer we verify a conjecture of Erdös and Simonovits concerning the number of copies of $C_4$ in a graph with $ex(q^2+q+1, C_4)+1$ edges. Further, we give a sufficient condition for the sum graph of…
Given a graph $H$, the Turán number $ex(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a recent conjecture of Conlon, Janzer, and Lee on the Turán numbers of bipartite graphs, which in turn yields further progress on a conjecture of Erdős and Simonovits. Let $s,t,k\geq 2$ be integers. Let $K_{s,t}^k$ denote the graph obtained from the complete bipartite graph $K_{s,t}$ by replacing each edge $uv$ in it with a path of length $k$ between $u$ and $v$ such that the $st$ replacing paths are internally disjoint. It follows from a general theorem of Buk…
Suppose that $R$ (red) and $B$ (blue) are two graphs on the same vertex set of size $n$, and $H$ is some graph with a red-blue coloring of its edges. How large can $R$ and $B$ be if $R\cup B$ does not contain a copy of $H$? Call the largest such integer $\mathrm{mex}(n, H)$. This problem was introduced by Diwan and Mubayi, who conjectured that (except for a few specific exceptions) when $H$ is a complete graph on $k+1$ vertices with any coloring of its edges $\mathrm{mex}(n,H)=\mathrm{ex}(n, K_{k+1})$. This conjecture generalizes Turán's theorem. Diwan and Mubayi also asked for an analogue of …
The rainbow Turán number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$, $\mathrm{ex}^*(n,C_{2k})=O(n^{1+1/k})$. This is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstraëte. We use the same method to prove several other conjectures in various topics. First, we prove that there exists a constant $c$ such that any properly edge-coloured $n$-vertex graph with more than $cn(\log n)^4$ edges contains a rainbow cycle. …
An $r$-uniform hypergraph is called an $r$-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear $r$-graph $H$ and a positive integer $n$, the linear Turán number $ex_L(n,H)$ is the maximum number of edges in a linear $r$-graph $G$ that does not contain $H$ as a subgraph. For each $\ell\geq 3$, let $C^r_\ell$ denote the $r$-uniform linear cycle of length $\ell$, which is an $r$-graph with edges $e_1,\ldots, e_\ell$ such that $\forall i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_\ell\cap e_1|=1$ and $e_i\cap e_j=\emptyset$ for all other pairs $\{i,j\}, i…
The \emph{Turán function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Turán and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine $h_F(n,q)$, the minimum number of copies of $F$ that a graph with $n$ vertices and $\ex(n,F)+q$ edges can have. We determine $h_F(n,q)$ asymptotically when $F$ is \emph{color-critical} (that is, $F$ contains an edge whose deletion reduces its chromatic number) and $q=o(n^2)$. Determining the exact value of $h_F(n,q)$ seems rather difficul…
Turán's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs. The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $γn^2$ edges has been intensively studied in the literature. In particular, Lovász and Simonovits conjectured in the 1970s that asymptotically the best possible lower bound is given by t…
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）
Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. Given a positive integer $n$ and a planar graph $H$, let $\mathcal{T}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains a subgraph isomorphic to $H$. The planar anti-Ramsey number of $H$, denoted $ar_{_\mathcal{P}}(n, H)$, is the maximum number of colors in an edge-coloring of a plane triangulation $T\in \mathcal{T}_n(H)$ such that $T$ contains no rainbow copy of $H$. Analogous to anti-Ramsey numbe…
Exact Turán number of the Fano plane in the $\ell_2$-norm
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）
On the Turán number of odd-ballooning of $3$-chromatic graphs
School of Mathematics, East China University of Science and Technology（华东理工大学数学学院）；School of Mathematics and Finance, Chuzhou University（滁州学院数学与金融学院）；School of Finance Business, Shanghai Normal University（上海师范大学商学院）
