Andrew Frohmader
For Turán's (3, 4)-conjecture, in the case of n = 3k+1 vertices, (.5)6^{k-1} non-isomorphic complexes are constructed that attain the conjecture. In the case of n = 3k+2 vertices, 6^{k-1} non-isomorphic complexes are constructed that attain the conjecture.
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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 $\mathrm{ex}(n,H)=O(n^{4/3+\varepsilon})$.
Hurwitz Stability of Generalized Turán Expressions of Polynomial Sequences
本文证明了Eulerian和Narayana多项式的Turán表达式稳定性猜想，推广了Turán表达式概念，给出广义稳定性判定准则，并应用于递推多项式序列。
We partly confirm a Brualdi-Solheid-Turán type conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. We confirm Nikiforov's Conjecture for all brooms and for a larger class of spiders. For our proofs we also obtain a new Turán type result which might turn out to be of independent interest.
On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Turán Theorem
本文通过引入覆盖理想方法，反驳了Mubayi关于超图Turán定理的多项式理想猜想，并给出了新的代数证明。
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…
Ramsey multiplicity and the Turán coloring
本文证明 Turán 染色对无限族图是唯一的极值染色，首次确定了 Burr-Rosta 猜想失效图的 Ramsey 多重性常数，并给出条件性的三色结果。
Two conjectured strengthenings of Turán's theorem
该研究提出两个强化Turán定理的两个谱图论猜想，在多类图上证明猜想成立并给出实验证据，相关猜想被列为谱图论重要未解决问题。
A survey of edge-spectral-Turán type problems in spectral graph theory: Results, conjectures and open problems
本文综述了谱图理论中边缘-谱-图论类型问题的研究成果，提出了相关猜想与开放问题。
On Turán exponents of bipartite graphs
本文证明了Erdős-Simonovits猜想对形如$2-2/(2k+1)$的有理数成立，并给出$θ$-图Turán数的上界，从而得到新的有理指数$r=7/5$，推进了相关猜想的研究。
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…
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 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…
Let $\mathcal{F}$ be a family of $r$-graphs. The Turán number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erdős Matching Conjecture shows that \[ ex_r(n,M_{k+1}^{(r)})= \max\left\{\binom{rk+r-1}{r},\binom{n}{r}-\binom{n-k}{r}\right\}, \] where $M_{k+1}^{(r)}$ represents the $r$-graph consisting of $k+1$ disjoint edges. Motivated by this conjecture, we consider the Turán problem for tight linear forests. A tight linear forest is an $r$-graph whose connected components are all tight paths or isolated verti…
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…
On the multicolor Turán conjecture for color-critical graphs
本文通过结合稳定性论证与新型图打包技术，证明了色临界图的多色Turán猜想在更广的k值范围内成立，显著改进了先前结果。
A step towards the Ramsey-Turán conjecture for $K_3$ and $K_6$
本文利用Szemerédi正则引理和稳定性论证，证明了对于 $K_3$ 和 $K_6$，Ramsey-Turán函数 $ρ(3,6,δ)$ 的上界为 $\frac{5}{12} + \frac{δ}{2}+ 2.1025δ^2$，为猜想提供了第一步证明。
Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let $M_t^r$ be the $r$-graph with $t$ pairwise disjoint edges. Given an $r$-graph $F$ and a positive integer $p\geq |V(F)|$, we define the {\em extension} of $F$, denoted by $H_{p}^{F}$ as follows: Label the vertices of $F$ as $v_1,\dots,v_{|V(F)|}$. Add new vertices $v_{|V(F)|+1},\dots,v_{p}$. For each pair of vertices $v_i,v_j, 1\le i<j \le p$ not contained in …
A conjecture of Erdős from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this bound holds for a large family of $r$-degenerate bipartite graphs, including all $r$-degenerate blow-ups of trees. Our results generalise many previously proven cases of the Erdős conjecture, including the related results of Füredi and Alon, Krivelevich and Sudakov. Our proof uses supersaturation and a random walk on an auxiliary graph.
A counterexample to the Fang--Lin conjecture: Edge and spectral extremality diverge near a Turán graph
本文构造反例 $F=K_1\vee\mu(K_3)$，证明存在边色临界图使得边极值与谱极值不一致，即无图能同时最大化边数和谱半径。
The family $Σ_r$ consists of all $r$-graphs with three edges $D_1,D_2,D_3$ such that $|D_1\cap D_2|=r-1$ and $D_1 \triangle D_2 \subseteq D_3$. A generalized triangle, $\mathcal{T}_r \in Σ_r$ is an $r$-graph on $\{1,2,\ldots,2r-1\}$ with three edges $D_1, D_2, D_3$, such that $D_1=\{1,2,\dots,r-1, r\}, D_2= \{1, 2, \dots, r-1, r+1 \}$ and $D_3 = \{r, r+1, \dots, 2r-1\}.$ Frankl and Füredi conjectured that for all $r\geq 4$, $ex(n,Σ_r) = ex(n,\mathcal{T}_r )$ for all sufficiently large $n$ and they also proved it for $r=3$. Later, Pikhurko showed that the conjecture holds for $r=4$. In this pap…
A Proof of the $(n,k,t)$-Conjectures
本研究针对$(n,k,t)$图的极值结构问题，证明了对任意t，最小$(n,k,t)$图都是团的不交并，推广了Turán定理，同时证实了Hoffman等人提出的两个相关猜想。
We prove the Schröder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schröder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schröder of Haglund (Haglund 2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another …
Recently, Domagoj Bradač, Oliver Janzer, Benny Sudakov and István Tomon have proved that the Turán number of $2$-dimensional grids is $Θ(n^{3/2})$, or more general, $\mathrm{ex}\left(n,T\square{P}\right)=Θ(n^{3/2})$, where $T$ is a non-trivial tree, $P$ is a non-trivial path, and $T\square{P}$ denotes the Cartesian product. In their proof, they exhibited a novel way of using the tensor power trick, which has lots of potential in Turán type problems. By the end of their proof, they conjectured that $\mathrm{ex}\left(n,T\square{R}\right)=Θ(n^{3/2})$ for non-trivial trees $T$ and $R$. This paper …
Turán Densities for Daisies and Hypercubes
本文推翻了 $r$-daisy 的 Turán 密度随 $r$ 趋于无穷而趋于零的猜想，并调整构造进一步推翻了关于超立方体 Turán 密度的民间猜想，证明了其渐近密度的更紧上界。
Counterexamples to a treewidth conjecture on generalized Turán problems
本研究针对广义Turán问题中关于色数与树宽的猜想构造反例，证明对所有r≥3存在满足色数和树宽均为r的图F_r，使得ex(n,K_r,F_r)为o(n^{r-1})，证否了相关猜想与综述中的对应结论。
Given an $r$-graph $H$ on $h$ vertices, and a family $\mathcal{F}$ of forbidden subgraphs, we define $\ex_{H}(n, \mathcal{F})$ to be the maximum number of induced copies of $H$ in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Then the \emph{Turán $H$-density} of $\mathcal{F}$ is the limit \[π_{H}(\mathcal{F})= \lim_{n\rightarrow \infty}\ex_{H}(n, \mathcal{F})/\binom{n}{h}. \] This generalises the notions of \emph{Turán density} (when $H$ is an $r$-edge), and \emph{inducibility} (when $\mathcal{F}$ is empty). Although problems of this kind have received some attention, very few results are k…
On the "second" Kahn--Kalai Conjecture: cliques, cycles, and trees
本文证明了蕴含第二个Kahn--Kalai猜想的随机图陈述的几个简单情形，并希望所引入的思想能推动该自然问题的进一步研究。
The Turán number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erdős-Sós conjecture, and conditional on its truth we determine ex(n;H) when H is an equibipartite forest, for appropriately large n.
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. …
The Turán number of a graph $H$, denoted by $ex(n, H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let $P_{\ell}$ denote the path on $\ell$ vertices, $S_{\ell-1}$ denote the star on $\ell$ vertices and $k_1P_{\ell}\cup k_2S_{\ell-1}$ denote the path-star forest with disjoint union of $k_1$ copies of $P_{\ell}$ and $k_2$ copies of $S_{\ell-1}$. In 2013, Lidický et al. first considered the Turán number of $k_1P_4\cup k_2S_3$ for sufficiently large $n$. In 2022, Zhang and Wang raised a conjecture about the Turán number of $k_1P_{2\ell}\cup k_2S_{2…
We show that there is a constant $c$ such that any 3-uniform hypergraph $\mathcal H$ with $n$ vertices and at least $cn^{5/2}$ edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of Kupavskii, Polyanskii, Tomon, and Zakharov. Furthermore, our work, combined with prior results, asymptotically determines the Turán number of all surfaces.
For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $δ(H)>0$ and $Δ(H)$ denote the minimum degree and maximum degree of $H$, respectively. We prove that for all $n$ sufficiently large, if $H$ is any graph of order $n$ with $Δ(H) \le \sqrt{n}/200$, then $ex(n,H)={{n-1} \choose 2}+δ(H)-1$. The condition on the maximum degree is tight up to a constant factor. This generalizes a classical result of Ore for the case $H=C_n$, and resolves, in a strong form, a conjecture of Glebov, Person, and Wep…
A positive square-energy strengthening of Turán's theorem
研究关于\(n\)顶点图\(G\)的正邻接特征值平方和\(s^+(G)\)与团数\(\omega(G)\)的关系，核心方法是将定理简化为双非负矩阵的莫茨金 - 施特劳斯不等式并通过局部逆概率估计证明，主要贡献是强化了威尔夫定理并解决猜想。
Biregular bipartite labeled multigraphs and perfect matchings in bipartite tensor products
本文研究双部张量积中的完美匹配问题，证明了某些特殊情况，并将猜想1转化为矩阵不等式，进一步证明了该不等式的特殊情况。
Heim, Neuhauser, and Tröger recently established some inequalities for MacMahon's plane partition function $\mathrm{PL}(n)$ that generalize known results for Euler's partition function $p(n)$. They also conjectured that $\mathrm{PL}(n)$ is log-concave for all $n\geq 12.$ We prove this conjecture. Moreover, for every $d\geq 1$, we prove their speculation that $\mathrm{PL}(n)$ satisfies the degree $d$ Turán inequality for sufficiently large $n$. The case where $d=2$ is the case of log-concavity.
The Turán inequalities and the higher order Turán inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. A real sequence $\{a_{n}\}$ is said to satisfy the Turán inequalities if for $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It is said to satisfy the higher order Turán inequalities if for $n\geq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0$. A sequence satisfying the Turán inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequ…
The Burning Number Conjecture Holds Asymptotically
本文证明了燃烧数猜想渐近成立，即对于任意$n$个顶点的图$G$，其燃烧数$b(G)$不超过$(1+o(1))\sqrt n$，从而在渐近意义上解决了该猜想。
Lights Out! is a game played on a $5 \times 5$ grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration of lit lights and reach a state where all lights are out. Two conjectures posed in a recently published paper about Lights Out! on Cartesian products of graphs are resolved.
On the Erdős-Turán Conjecture and the growth of $B_{2}[g]$ sequences
本文研究Erdős-Turán猜想中B₂[g]序列的加法基性质与增长密度；通过构造满足两两和表示数受限的序列，证明在g>1/ε时其可带小项三项表示所有充分大整数，并将计数下界改进为≫x^{g/(2g+1)}。
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