On the asymptotic value of the choice number of complete multi-partite graphs

Minimum vertex degree thresholds for tiling complete 3-partite 3-graphs

Given positive integers $a\leq b \leq c$, let $K_{a,b,c}$ be the complete 3-partite 3-uniform hypergraph with three parts of sizes $a,b,c$. Let $H$ be a 3-uniform hypergraph on $n$ vertices where $n$ is divisible by $a+b+c$. We asymptotically determine the minimum vertex degree of $H$ that guarantees a perfect $K_{a, b, c}$-tiling, that is, a spanning subgraph of $H$ consisting of vertex-disjoint copies of $K_{a, b, c}$. This partially answers a question of Mycroft, who proved an analogous result with respect to codegree for $r$-uniform hypergraphs for all $r\ge 3$. Our proof uses a lattice-ba…

完全S-部图中最大的2-正则子图

Largest $2$-regular Subgraphs in complete $S$-partite Graphs

研究完全S-部图中最大2-正则子图问题，提出通过整数线性规划求解，并证明该问题可在O(|V(S)|^3)时间内高效解决，同时通过模拟验证随机S-部图中存在与完整图相同阶数的最大2-正则子图。

准随机k-部超图中的铺砌

Tilings in quasi-random $k$-partite hypergraphs

本文研究了准随机k-部超图中F-因子的存在性，证明了当p>1/2且满足部最小共度条件时，超图中存在F-因子，并展示了1/2是该阈值的最优界。

组合零点定理与完全 r-部 r-一致超图的 Turán 数

Combinatorial Nullstellensatz and Turán numbers of complete $r$-partite $r$-uniform hypergraphs

本文利用 Lasoń 推广的组合零点定理框架构造完全 r-部 r-一致超图 Turán 数的下界，并给出 Erdős 盒子问题的简短显式构造，证明当 r≤4 时渐近匹配已知最佳界。

On the choice number of complete multipartite graphs with part size four

Let $\mathrm{ch}(G)$ denote the choice number of a graph $G$, and let $K_{s*k}$ be the complete $k$-partite graph with $s$ vertices in each part. Erdős, Rubin, and Taylor showed that $\mathrm{ch}( K_{2*k})=k$, and suggested the problem of determining the choice number of $K_{s*k}.$ The first author established $\mathrm{ch}( K_{3*k})=\left\lceil \frac{4k-1}{3}\right\rceil$. Here we prove $\mathrm{ch} (K_{4*k})=\left\lceil \frac{3k-1}{2}\right\rceil$.

Codegree conditions for tiling complete $k$-partite $k$-graphs and loose cycles

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 …

完全图和平衡完全r部图上的错位完美匹配

Deranged Perfect Matchings on complete graph and balanced complete r-partite graph

研究了完全图和平衡完全r部图中随机完美匹配与稀疏子图的交集分布，证明了其渐近独立泊松分布性质，扩展了近期多项研究成果。

Codegree conditions for tilling balanced complete $3$-partite $3$-graphs and generalized 4-cycles

Given two $k$-graphs $F$ and $H$, a perfect $F$-tiling (also called an $F$-factor) in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. Let $t_{k-1}(n, F)$ be the smallest integer $t$ such that every $k$-graph $H$ on $n$ vertices with minimum codegree at least $t$ contains a perfect $F$-tiling. Mycroft (JCTA, 2016) determined the asymptotic values of $t_{k-1}(n, F)$ for $k$-partite $k$-graphs $F$. Mycroft also conjectured that the error terms $o(n)$ in $t_{k-1}(n, F)$ can be replaced by a constant that depends only on $F$. In this paper, we improve the er…

三部3-一致超图中完美匹配的谱条件

A spectral condition for perfect matchings in 3-partite 3-graphs

针对三部3-一致超图，证明了若每个顶点的链接图谱半径大于(√2/2+α)n，则存在完美匹配，且该条件渐近最优。

The Zarankiewicz problem in 3-partite graphs

Let $F$ be a graph, $k \geq 2$ be an integer, and write $\mathrm{ex}_{ χ\leq k } (n , F)$ for the maximum number of edges in an $n$-vertex graph that is $k$-partite and has no subgraph isomorphic to $F$. The function $\mathrm{ex}_{ χ\leq 2} ( n , F)$ has been studied by many researchers. Finding $\mathrm{ex}_{ χ\leq 2} (n , K_{s,t})$ is a special case of the Zarankiewicz problem. We prove an analogue of the Kövári-Sós-Turán Theorem for 3-partite graphs by showing \[ \mathrm{ex}_{ χ\leq 3} (n , K_{s,t} ) \leq \left( \frac{1}{3} \right)^{1 - 1/s} \left( \frac{ t - 1}{2} + o(1) \right)^{1/s} n^{2…

An asymptotic multipartite Kühn-Osthus theorem

In this paper we prove an asymptotic multipartite version of a well-known theorem of Kühn and Osthus by establishing, for any graph $H$ with chromatic number $r$, the asymptotic multipartite minimum degree threshold which ensures that a large $r$-partite graph $G$ admits a perfect $H$-tiling. We also give the threshold for an $H$-tiling covering all but a linear number of vertices of $G$, in a multipartite analogue of results of Komlós and of Shokoufandeh and Zhao.

环的部分饱和数

Partite saturation number of cycles

本文研究了完全$k$-部图$K_k^n$中长度为$\ell$的环的饱和数，给出了除$(4,4)$外所有$\ell\geq4,k\geq2$的渐近紧界，并确定了若干情形的精确值。

k部图中无大空洞的横贯铺砌

Transversal tilings in k-partite graphs without large holes

该研究证明了在k部图中，满足特定度数和空洞条件的图存在横贯因子，扩展了相关领域的最新成果。

关于Füredi的t-部链接问题的注记

A note on the $t$-partite link problem of Füredi

研究了3-图中每个顶点的链接图为t-部图时的链接密度上限，证明了上限为1−t⁻¹−t⁻²/12，并结合Goldwasser的构造确定了t−1为素数幂时的渐进行为。

Asymptotic multipartite version of the Alon-Yuster theorem

In this paper, we prove the asymptotic multipartite version of the Alon-Yuster theorem, which is a generalization of the Hajnal-Szemerédi theorem: If $k\geq 3$ is an integer, $H$ is a $k$-colorable graph and $γ>0$ is fixed, then, for every sufficiently large $n$, where $|V(H)|$ divides $n$, and for every balanced $k$-partite graph $G$ on $kn$ vertices with each of its corresponding $\binom{k}{2}$ bipartite subgraphs having minimum degree at least $(k-1)n/k+γn$, $G$ has a subgraph consisting of $kn/|V(H)|$ vertex-disjoint copies of $H$. The proof uses the Regularity method together with linear …

Bipartite Turán problems for ordered graphs

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-\…

Turán Problems for Vertex-disjoint Cliques in Multi-partite Hypergraphs

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-…

Off-diagonal book Ramsey numbers

The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n^{(k)}, B_n^{(k)})$. Here we consider the natural off-diagonal variant $r(B_{cn}^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$. In this more general setting, we show that an interesting dichotomy emerges: for very small $c$, a simple $k$-partite construction dictates the Ramsey function and all nearly-extremal colorings are close to being $k$-partite, while, for $c$ bounded away from $0$, random colorings of an appropriate density ar…

极值数与Sidorenko猜想

Extremal numbers and Sidorenko's conjecture

本文针对Sidorenko猜想在超图上的反例，证明若猜想对某r部r均匀超图不成立，则可改进其极值数的概率删除法下界，并给出包括含松散三角形线性超图及3部3均匀紧环在内的新反例。

Decomposing the Complete $r$-Graph

Let $f_r(n)$ be the minimum number of complete $r$-partite $r$-graphs needed to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. Graham and Pollak showed that $f_2(n) = n-1$. An easy construction shows that $f_r(n)\le (1-o(1))\binom{n}{\lfloor r/2\rfloor}$ and it has been unknown if this upper bound is asymptotically sharp. In this paper we show that $f_r(n)\le (\frac{14}{15}+o(1))\binom{n}{r/2}$ for each even $r\ge 4$.

Crossing numbers of complete tripartite and balanced complete multipartite graphs

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr'(K_{n_1,n_2})\le Z(n_1,n_2):= n_1/2*(n_1-1)/2*n_2/2*(n_2-1)/2. We define an analogous bound A(n_1,n_2,n_3) for the complete tripartite graph K_{n_1,n_2,n_3}, and prove that cr'(K_{n_1,n_2,n_3})\le A({n_1,n_2,n_3}). We also show that for n large enough, 0.973 A(n,n,n) \le cr'(K_{n…

Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from any set of lists associated to the edges of $G$ as long as the size of every list is at least $Δ+CΔ^{\frac{1}{2}}(\logΔ)^4$, where $Δ$ is the maximum degree of $G$ and $C$ is a constant. The proof is probabilistic. The same is true in the environment of total colourings.

Choosability with separation of complete multipartite graphs and hypergraphs

For a hypergraph G and a positive integer s, let χ_{\ell} (G,s) be the minimum value of l such that G is L-colorable from every list L with |L(v)|=l for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This parameter was studied by Kratochvíl, Tuza and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of χ_{\ell} (G,s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs.

Choice number of complete multipartite graphs $K_{3*3,2*(k-5),1*2}$ and $K_{4,3*2,2*(k-6),1*3}$

A graph $G$ is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=χ(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|\leq 2χ(G)+1$ is chromatic-choosable. Since each $k$-chromatic graph is a subgraph of a complete $k$-partite graph, we see that Ohba's conjecture is true if and only if it is true for every complete multipartite graph. However, the only complete multipartite graphs for which Ohba's conjecture has been verified are: $K_{3*2,2*(k-3),1}$, $K_{3,2*(k-1)}$, $K_{s+3,2*(k-s-1),1*s}$, $K_{4,3,2*(k-4),1*2}$, and $K_{5,3,…

On two problems in Ramsey-Turán theory

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…

关于部分选择有限的划分的偏差与渐近性

On the biases and asymptotics of partitions with finite choices of parts

本文研究整数划分中来自不同集合的部分数量比较所产生的偏差，在部分集合有限时给出相关划分数之比的渐近公式，并对无限算术级数情形提出猜想。

ℓ₂范数下的四面体猜想

Tetrahedron Conjecture in the $\ell_2$-norm

本文证实了Balogh-Clemen-Lidický提出的ℓ₂范数下Turán四面体猜想的唯一极值性，通过顶点着色图的Mantel定理和Simonovits稳定性方法的新程序完成证明，相关策略对其他极值问题有潜在价值。

Cliques in dense inhomogeneous random graphs

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of G(n,W) for each fixed n, and give examples of graphons for which G(n,W) exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent i…

The asymptotic induced matching number of hypergraphs: balanced binary strings

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith-Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to ques…

关于偶数 $t$ 时 $ex(n,n,n,K_{2,t})$ 的精确渐近性

On the Precise Asymptotics of $ex(n,n,n,K_{2,t})$ for even $t$

本文证明了偶数 $t$ 时三部图不含 $K_{2,t}$ 的最大边数渐近下界，并结合已有工作确定其精确极限为 $\frac{3\sqrt{t-1}}{\sqrt{2}} n^{3/2}$。

On the abstract chromatic number and its computability for finitely axiomatizable theories

The celebrated Erdős--Stone--Simonovits theorem characterizes the asymptotic maximum edge density in $\mathcal{F}$-free graphs as $1 - 1/(χ(\mathcal{F})-1) + o(1)$, where $χ(\mathcal{F})$ is the minimum chromatic number of a graph in $\mathcal{F}$. In Examples 25 and 31 of [L. N. Coregliano and A. A. Razborov. Semantic limits of dense combinatorial objects. Uspekhi Mat. Nauk, 75(4(454)):45-152, 2020], it was shown that this result can be extended to the general setting of graphs with extra structure: the maximum asymptotic density of a graph with extra structure without some induced subgraphs …

Erdős-Rényi 超图的平衡着色

Balanced colorings of Erdős-Rényi hypergraphs

本文研究了稀疏 $r$-一致 $r$-部 Erdős-Rényi 超图的平衡着色问题，确定了其平衡色数的渐近行为，并将 Feige 和 Kogan 关于二部图的结果推广到了更一般的超图情形。

Smirnov词与完全多部图中哈密顿圈的双射

Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs

建立平衡字母重数的Smirnov词与完全m部图K_{n,n,...,n}中哈密顿路径的双射，导出哈密顿圈数的闭式容斥公式，并推广到非均匀情形K_{n1,n2,...,nm}，给出渐近分析。

On-line list colouring of random graphs

In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C in [2,4], depending on the range of d. Also, for d=O(1…