Dániel Gerbner
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.
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Let $H$ be a graph and $t\geq s\geq 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $λ(G) = O\left(n^{1-1/s}\right)$ where $λ(G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.
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.
Generalized Turán results for disjoint copies of degenerate graphs
本文扩展了广义Turán数研究，针对禁止图$(t+1)K_{a,b}$和$(t+1)C_{2k}$，建立了与已知结果同阶的精确匹配上下界。
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…
In this paper, we consider the Turán problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\}, \{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Turán densities of some small $\{1,3\}$-hypergraphs.
Given a hypergraph $\mathcal{H}$ and a graph $G$, we say that $\mathcal{H}$ is a \textit{Berge}-$G$ if there is a bijection between the hyperedges of $\mathcal{H}$ and the edges of $G$ such that each hyperedge contains its image. We denote by $ex_k(n,\text{Berge-}F)$ the largest number of hyperedges in a $k$-uniform Berge-$F$-free graph. Let $ex(n,H,F)$ denote the largest number of copies of $H$ in $n$-vertex $F$-free graphs. It is known that $ex(n,K_k,F)\le ex_k(n,\text{Berge-}F)\le ex(n,K_k,F)+ex(n,F)$, thus if $χ(F)>r$, then $ex_k(n,\text{Berge-}F)=(1+o(1)) ex(n,K_k,F)$. We conjecture that …
A note on the exact partition polytope of Frieze and Teng
本文指出Frieze和Teng提出的精确划分问题的整数线性规划松弛可能产生退化多面体，并研究了退化条件，给出了最小退化实例及等价非退化实例，同时说明可通过预处理避免退化。
For fixed integers $r>k\ge 2,e\ge 3$, let $f_r(n,er-(e-1)k,e)$ be the maximum number of edges in an $r$-uniform hypergraph in which the union of any $e$ distinct edges contains at least $er-(e-1)k+1$ vertices. A classical result of Brown, Erdős and Sós in 1973 showed that $f_r(n,er-(e-1)k,e)=Θ(n^k).$ The degenerate Turán density is defined to be the limit (if it exists) $$π(r,k,e):=\lim_{n\rightarrow\infty}\frac{f_r(n,er-(e-1)k,e)}{n^k}.$$ Extending a recent result of Glock for the special case of $r=3,k=2,e=3$, we show that $$π(r,2,3):=\lim_{n\rightarrow\infty}\frac{f_r(n,3r-4,3)}{n^2}=\frac{…
For fixed integers $r\ge 3, e\ge 3$, and $v\ge r+1$, let $f_r(n,v,e)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph in which the union of arbitrary $e$ distinct edges contains at least $v+1$ vertices. In 1973, Brown, Erdős and Sós proved that $f_r(n,er-(e-1)k,e)=Θ(n^k)$ and conjectured that the limit $\lim_{n\rightarrow\infty}\frac{f_3(n,e+2,e)}{n^2}$ always exists for all fixed integers $e\ge 3$. In 2020 Shangguan and Tamo conjectured that the limit $\lim_{n\rightarrow\infty}\frac{f_r(n,er-(e-1)k,e)}{n^k}$ always exists for all fixed integers $r>k\ge 2$ and $e\ge …
On non-degenerate Turán problems for expansions
本文研究图的r一致展开的Turán问题，证明其极值数等于无F图中r团的最大数量加上低阶项，并借助结构定理对部分图类给出精确结果。
Intervals of hypergraph Turán densities
本文证明了对每个整数r≥3，r-图族的Turán密度集合包含非退化区间，从而回答了Frankl等人2007年的问题，并证明了Hausdorff维数为1。
For every integer $t \ge 0$, denote by $F_5^t$ the hypergraph on vertex set $\{1,2,\ldots, 5+t\}$ with hyperedges $\{123,124\} \cup \{34k : 5 \le k \le 5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t\ge 0$ and sufficiently large $n$ and characterize the extremal $F_5^t$-free hypergraphs. In particular, if $n$ satisfies certain divisibility conditions, then the extremal $F_5^t$-free hypergraphs are exactly the balanced complete tripartite hypergraphs with additional hyperedges inside each of the three parts $(V_1,V_2,V_3)$ in the partition; each part $V_i$ spans a $(|V_i|,3,2,t)$-desig…
Given graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. Stability refers to the usual phenomenon that if an $n$-vertex $F$-free graph $G$ contains almost $\mathrm{ex}(n,H,F)$ copies of $H$, than $G$ is in some sense similar to some extremal graph. We obtain new stability results for generalized Turán problems and derive several new exact results.
Exact generalized Turán number of vertex-disjoint paths of length two
研究确定长度为二的顶点不相交路径的广义图兰数及极值图，核心方法是结合克鲁斯卡尔 - 卡托纳定理的洛瓦兹形式与离散凸性论证。
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