对于正整数k和图H，k-颜色诱导大小拉姆齐数R_ind(H,k)是使得存在一个边数为m的图G，使得对于任何k-边着色的G，G包含H的一个单色诱导子图的最小整数m。对于图H的边集E(H)和函数σ:E(H)→N，通过将每条边e替换为长度为σ(e)的路径得到细分H^σ。证明对于所有整数k,D≥2，存在常数c=c(k,D)，使得对于任何最大度为D的图H，以及H^σ的细分，其中σ(e) > c log_D n对于所有e∈E(H)，则R_ind(H^σ,k)=e^{O(k log k)} D^9 (log D) n。若每个σ(e)为偶数且大于c log_D n，则该界改进为R_ind(H^σ,k)=O(k^{342} (log k)^9 D^9 log D )n。此外，还找到了长细分非诱导大小拉姆齐数的改进界。
英文摘要：
For a positive integer $k$ and a graph $H$, the $k$-color induced size-Ramsey number $\hat{R}_{\mathrm{ind}}(H, k)$ is the minimum integer $m$ for which there exists a graph $G$ with $m$ edges such that for every $k$-edge coloring of $G$, the graph $G$ contains a monochromatic copy of $H$ as an induced subgraph. For a graph $H$ with the edge set $E(H)$ and a function $σ:E(H)\to \mathbb{N}$, the subdivision $H^σ$ is obtained by replacing each $e \in E(H)$ with a path of length $σ(e)$. We prove that for all integers $k,\, D\geq 2$, there exists a constant $c=c(k, D)$ such that the following holds. Let $ H $ be any graph with maximum degree $D$ and let $H^σ$ be a subdivision of $H$ with $σ(e) > c \log_D n $ for every $e \in E(H)$, where $n$ is the order of $H^σ$. Then, $\hat{R}_{\mathrm{ind}}(H^σ,k)=e^{O(k\log k)} D^{9}(\log D)\, n$. If each $σ(e)$ is even and larger than $c \log_D n$, this bound improves to $\hat{R}_{\mathrm{ind}}(H^σ,k)=O(k^{342} (\log k)^9D^{9} \log D )n$. We also find improved bounds for the non-induced size-Ramsey number of long subdivisions.
In this paper, for small uniformities, we determine the order of magnitude of the multicolor Ramsey numbers for Berge cycles of length $4$, $5$, $6$, $7$, $10$, or $11$. Our result follows from a more general setup which can be applied to other hypergraph Ramsey problems. Using this, we additionally determine the order of magnitude of the multicolor Ramsey number for Berge-$K_{a,b}$ for certain $a$, $b$, and uniformities.
A recent breakthrough of Conlon and Ferber yielded an exponential improvement on the lower bounds for multicolor diagonal Ramsey numbers. In this note, we modify their construction and obtain improved bounds for more than three colors.
The multicolor Ramsey number problem asks, for each pair of natural numbers $\ell$ and $t$, for the largest $\ell$-coloring of a complete graph with no monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. Graphs useful for this construction are exactly those relevant for a problem of Erdős on graphs with no large cliques and few large independent sets. We also make some basic observations about thi…
We show that if we color the hyperedges of the complete $3$-uniform complete graph on $2n+\sqrt{18n+1}+2$ vertices with $n$ colors, then one of the color classes contains a loose path of length three.
We show that there exists an absolute constant $A$ such that for each $k\ge2$ and every coloring of the edges of the complete $k$-uniform hypergraph on $ Ar$ vertices with $r$ colors, one of the color classes contains a loose path of length three.
Let $r_k(C_{2m+1})$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge2$, \[r_k(C_{2m+1})<c^{k}\sqrt{k!}\] for all sufficiently large $k$, where $c=c(m)>0$ is a constant. This improves an old result by Bondy and Erdős (Ramsey numbers for cycles in graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).
The list Ramsey number $R_{\ell}(H,k)$, recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical Ramsey number. They showed that if $H$ is a fixed $r$-uniform hypergraph that is not $r$-partite and the number of colors $k$ goes to infinity, $e^{Ω(\sqrt{k})} \le R_{\ell} (H,k) \le e^{O(k)}$. We prove that $R_{\ell}(H,k) = e^{Θ(k)}$ if and only if $H$ is not $r$-partite.
Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $ \mathcal{L}$ be the loose 3-uniform path with 3 edges and $ \mathcal{M}$ denote the messy 3-uniform path with 3 edges; that is, let $\mathcal{L} = \{abc, cde, efg\}$ and $\mathcal{M} = \{ abc, bcd, def\}$. In this note we prove $ r_k(\mathcal{L}) < 1.54k$ and $ r_k(\mathcal{M}) < 1.6k$ for $k$ sufficiently large.
We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by Sárközy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a l…
The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers tha…
The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every $n$-vertex graph with at least $\varepsilon n^2$ edges contains a $1$-subdivision of the complete graph on $c_{\varepsilon}\sqrt{n}$ vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least $(2+o(1))k^2$ vertices contains th…
A weakly optimal $K_s$-free $(n,d,λ)$-graph is a $d$-regular $K_s$-free graph on $n$ vertices with $d=Θ(n^{1-α})$ and spectral expansion $λ=Θ(n^{1-(s-1)α})$, for some fixed $α>0$. Such a graph is called optimal if additionally $α= \frac{1}{2s-3}$. We prove that if $s_{1},\ldots,s_{k}\ge3$ are fixed positive integers and weakly optimal $K_{s_{i}}$-free pseudorandom graphs exist for each $1\le i\le k$, then the multicolor Ramsey numbers satisfy \[ Ω\Big(\frac{t^{S+1}}{\log^{2S}t}\Big)\le r(s_{1},\ldots,s_{k},t)\le O\Big(\frac{t^{S+1}}{\log^{S}t}\Big), \] as $t\rightarrow\infty$, where $S=\sum_{i…
A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. We study Ramsey-type problems in Gallai colorings. Given an integer $k\ge1$ and a graph $H$, the Gallai-Ramsey number $GR_k(H)$ is the least positive integer $n$ such that every Gallai $k$-coloring of the complete graph on $n$ vertices contains a monochromatic copy of $H$. It turns out that $GR_k(H)$ is more well-behaved than the classical Ramsey number $R_k(H)$. However, finding exact values of $GR_k (H)$ is far from trivial. In this …
For integers $k,r\geq 2$, the diagonal Ramsey number $R_r(k)$ is the minimum $N\in\mathbb{N}$ such that every $r$-coloring of the edges of a complete graph on $N$ vertices yields on a monochromatic subgraph on $k$ vertices. Here we make a careful effort of extracting explicit upper bounds for $R_r(k)$ from the pigeonhole principle alone. Our main term improves on previously documented explicit bounds for $r\geq 3$, and we also consider an often ignored secondary term, which allows us to subtract a uniformly bounded below positive proportion of the main term. Asymptotically, we give a self-cont…
The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest number of edges a graph $G$ can have, such that for every edge-coloring of $G$ with $r$ colors there exists a monochromatic copy of $H$ in $G$. For a graph $H$, we denote by $H^q$ the graph obtained from $H$ by subdividing its edges with $q{-}1$ vertices each. In a recent paper of Kohayakawa, Retter and R{ö}dl, it is shown that for all constant integers $q,r\geq 2$ and every graph $H$ on $n$ vertices and of bounded maximum degree, the $r$-size-Ramsey number of $H^q$ is at most $(\log n)^{20(q-1)}n^{1+1/q}$, for $n$ large …
Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In this short note, we study the lower bound for $R_k(H;K_m)$ when $H$ is $C_5$ or $C_7$, respectively. We show that \begin{equation*} R_{k}(C_5;K_m)=Ω(m^{\frac{3k}{8}+1}/(\log{m})^{\frac{3k}{8}+1}), \end{equation*} and \begin{equation*} R_{k}(C_7;K_m)=Ω(m^{\frac{2k}{9}+1}/(\log{m})^{\frac{2k}{9}+1}), \end{equation*} for fixed positive integer $k$ and $m\rightarro…
Given a positive integer $ r $, the $ r $-color size-Ramsey number of a graph $ H $, denoted by $ \hat{R}(H, r) $, is the smallest integer $ m $ for which there exists a graph $ G $ with $ m $ edges such that, in any edge coloring of $ G $ with $ r $ colors, $G$ contains a monochromatic copy of $ H $. Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle $ C_n $ is linear in $ n $ i.e. $ \hat{R}(C_n, r) \leq c_rn $, for some constant $ c_r $. Their proof, however, is based on the Szemerédi's regularity lemma and so no specific constant $ c_r $ is known. Javadi, Khoeini, O…
Multicolor Ramsey and list Ramsey numbers for star-like trees
School of Mathematical Sciences, Huaqiao University（华侨大学数学科学学院）；Academy of Plateau Science and Sustainability and School of Mathematics and Statistics, Qinghai Normal University（青海师范大学高原科学与可持续发展研究院及数学与统计学院）；Department of Mathematics and Statistics, Wright State University（赖特州立大学数学与统计系）
Let the integers $1,\ldots,n$ be assigned colors. Szemerédi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing totally multicolored arithmetic progressions of length 3. Let $f(n)$ be the smallest integer $k$ such that there is a coloring of $\{1, \ldots, n\}$ without totally multicolored arithmetic progressions of length three and such that each color appears on at most $k$ integers. We provide an exact value for $f(n)$ when $n$ is sufficiently large, and all extremal…
For given graphs $G_{1}, G_{2}, ... , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, ... , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with $k$ colors, then it always contains a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. The bipartite Ramsey number $b(G_1, \cdots, G_k)$ is the least positive integer $b$ such that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a monochromatic copy of bipartite $G_i$ in the $i$-th color, for some $i$, $1 \le i \le k$. Th…
A subposet $Q'$ of a poset $Q$ is a copy of a poset $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x\leq y$ in $P$ iff $f(x)\leq f(y)$ in $Q'$. For posets $P, P'$, let the poset Ramsey number $R(P,P')$ be the smallest $N$ such that no matter how the elements of the Boolean lattice $Q_N$ are colored red and blue, there is a copy of $P$ with all red elements or a copy of $P'$ with all blue elements. We provide some general bounds on $R(P,P')$ and focus on the situation when $P$ and $P'$ are both Boolean lattices. In addition, we give asymptotically tight bounds for t…
The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the $1$-subdivision of the $k$-vertex transitive tournament $H_k$ satisfies $\vec{r}(H_k)= O(k^2\log\log k)$. This is tight up to multiplicative $\log\log k$-term. We also show that if $T$ is an $n$-vertex tournament with $Δ^+(T)-δ^+(T)= O(n/k) - k^2$, then $T$ contains a $1$-subdivision of $\vec{K}_k$, a complete $k$-vertex digraph with all possible $k(k-1)$ arcs. This is also tight up to multiplicative constant.
Schrijver Number Quasi-Tensorization and Multicolor Ramsey Bounds via Robust OR Polynomials
Georgia Institute of Technology, School of Computer Science（佐治亚理工学院计算机科学学院）；Princeton University, Operations Research and Financial Engineering（普林斯顿大学运筹学与金融工程学）
We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of $K_n$ where each color is well-represented. Let $χ$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $χ$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $χ$ of $K_n$ contain a subgraph isomorphic…
In this paper, we prove that the multicolored Ramsey number $R(G_1,\dots,G_n,K_{n_1},\dots,K_{n_r})$ is at least $(γ-1)(κ-1)+1$ for arbitrary connected graphs $G_1,\dots,G_n$ and $n_1,\dots,n_r\in\mathbb{N}$, where $γ=R(G_1,\dots,G_n)$ and $κ=R(K_{n_1},\dots,K_{n_r})$. Erd\H os at al. conjectured that $R(C_n,K_l)=(n-1)(l-1)+1$ for every $n\geq l\geq 3$ except for $n=l=3$. Nikiforov proved this conjecture for $n\geq 4l+2$. Using the above bound, we derive the following generalization of this result. $R(C_n,K_{n_1},\dots,K_{n_r})=(n-1)(κ-1)+1$, where $κ=R(K_{n_1},\dots,K_{n_r})$ and $n\geq 4κ+2$…
