In this paper we study Turán and Ramsey numbers in linear triple systems, defined as $3$-uniform hypergraphs in which any two triples intersect in at most one vertex.
A famous result of Ruzsa and Szemerédi is that for any fixed $c>0$ and large enough $n$ the following Turán-type theorem holds. If a linear triple system on $n$ vertices has at least $cn^2$ edges then it contains a {\em triangle}: three pairwise intersecting triples without a common vertex. In this paper we extend this result from triangles to other triple systems, called {\em $s$-configurations}. The main tool is a generalization of the induced matching lemma from $aba$-patterns to more general ones.
We slightly generalize $s$-configurations to {\em extended $s$-configurations}. For these we cannot prove the corresponding Turán-type theorem, but we prove that they have the weaker, Ramsey property: they can be found in any $t$-coloring of the blocks of any sufficiently large Steiner triple system. Using this, we show that all unavoidable configurations with at most 5 blocks, except possibly the ones containing the sail $C_{15}$ (configuration with blocks 123, 345, 561 and 147), are $t$-Ramsey for any $t\geq 1$. The most interesting one among them is the {\em wicket}, $D_4$, formed by three rows and two columns of a $3\times 3$ point matrix. In fact, the wicket is $1$-Ramsey in a very strong sense: all Steiner triple systems except the Fano plane must contain a wicket.
We determine Turán numbers for the family of 3-uniform minimal paths of length four \emph{for all $n$}. We also establish the second and third order Turán numbers and use them to compute the corresponding Ramsey numbers for up to four colors.
We introduce and study a new type of Ramsey-Turan problems, a typical example of which is the following one: let c>0 and G be a graph of sufficiently large order n with minimum degree >3n/4. If the edges of G are colored in blue or red, then for every 3<k<(1/8-c)n, there exists a monochromatic cycle of length k.
We study Turán and Ramsey-type problems on edge-colored graphs. An edge-colored graph is called {\em $\varepsilon$-balanced} if each color class contains at least an $\varepsilon$-proportion of its edges. Given a family $\mathcal{F}$ of edge-colored graphs, the Ramsey function $R(\varepsilon, \mathcal{F})$ is the smallest $n$ for which any $\varepsilon$-balanced $K_n$ must contain a copy of an $F\in\mathcal{F}$, and the Turán function $\mathrm{ex}(\varepsilon, n, \mathcal{F})$ is the maximum number of edges in an $n$-vertex $\varepsilon$-balanced graph which avoids all of $\mathcal{F}$. In thi…
Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\le 7$. The proof of this result relies on a careful analysis of the Turán numbers for $P$. In this paper, we refine this analysis further and compute, for all $n$, the third and fourth order Turán numbers for $P$. With the help of the former, we confirm the formula $R(P;r)=r+6$ for $r\in\{8,9\}$.
Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\le 9$. The proof of this result relies on a careful analysis of the Turán numbers for $P$. In this paper, we refine this analysis further and compute the fifth order Turán number for $P$, for all $n$. Using this number for $n=16$, we confirm the formula $R(P;10)=16$.
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…
Let $n,r,k,s$ be positive integers with $n,k\ge 2$. The generalized Ramsey number $R(n,r;k,s)$ is the smallest positive integer $p$ such that for every graph $G$ of order $p$, either $G$ contains a subgraph induced by $n$ vertices with at most $r-1$ edges, or the complement $\bar G$ of $G$ contains a subgraph induced by $k$ vertices with at most $s-1$ edges. In this paper we completely determine $R(n,n(n-1)/2-r;k,1)$ for $n\ge 4$ and $r\le n-2$, and pose several conjectures on Ramsey numbers.
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 …
We say that a subgraph $F$ of a graph $G$ is singular if the degrees $d_G(v)$ are all equal or all distinct for the vertices $v\in V(F)$. The singular Ramsey number Rs$(F)$ is the smallest positive integer $n$ such that, for every $m\geq n$, in every edge 2-coloring of $K_m$, at least one of the color classes contains $F$ as a singular subgraph. In a similar flavor, the singular Turán number Ts$(n,F)$ is defined as the maximum number of edges in a graph of order $n$, which does not contain $F$ as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop me…
Let r be an integer, f(n) a function, and H a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi, the r-Ramsey-Turán number of H, RT_r(n, H, f(n)), is defined to be the maximum number of edges in an n-vertex, H-free graph G with α_r(G) <= f(n) where α_r(G) denotes the K_r-independence number of G. In this note, using isoperimetric properties of the high dimensional unit sphere, we construct graphs providing lower bounds for RT_r(n,K_{r+s},o(n)) for every 2 <= s <= r. These constructions are sharp for an infinite family of pairs of r and s. The only previous sharp construction was by Bollob…
Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Turán number $ex_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Turán numbers for $P$ under some additional restrictions. Most impo…
Given two graphs $G$ and $H$ with $H\subseteq G$ we consider the anti-Ramsey function $AR(G,H)$ which is the maximum number of colors in any edge-coloring of $G$ so that every copy of $H$ receives the same color on at least one pair of edges. The classical Turán function for a graph $G$ and family of graphs $\mathcal{F}$, written $ex(G,\mathcal{F})$, is defined as the maximum number of edges of a subgraph of $G$ not containing any member of $\mathcal{F}$. We show that there exists a constant $c>0$ so that $AR(K_n,K_{s,t})-ex(K_n,K_{s,t})<cn$ and $c$ depends only on $s$ and $t$, which implies $…
An important question in extremal graph theory raised by Vera T. Sós asks to determine for a given integer $t\ge 3$ and a given positive real number $δ$ the asymptotically supremal edge density $f_t(δ)$ that an $n$-vertex graph can have provided it contains neither a complete graph $K_t$ nor an independent set of size $δn$. Building upon recent work of Fox, Loh, and Zhao [The critical window for the classical Ramsey-Turán problem, Combinatorica 35 (2015), 435-476], we prove that if $δ$ is sufficiently small (in a sense depending on $t$), then \[ f_t(δ)= \begin{cases} \frac{3t-10}{3t-4}+δ-δ^2 &…
Given a graph $H$ and a function $f:\mathbb{Z}^+ \longrightarrow \mathbb{Z}^+ $, the Ramsey-Turán number of $H$ and $f$, denoted by $RT(n, H, f(n))$, is the maximum number of edges a graph $G$ on $n$ vertices can have, which does not contain $H$ as a subgraph and also does not contain a set of $f(n)$ independent vertices. Let $r$ be a positive integer. In 1969, Erdős and Sós proved that $RT(n,K_{2r+1},o(n))=\frac{n^2}{2}(1-\frac{1}{r})+o(n^2)$. Let $F_k(2r+1)$ denote the graph consisting of $k$ copies of complete graphs $K_{2r+1}$ sharing exactly one vertex. In this paper, we show that $RT(n,F…
The first application of Szemerédi's powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollobás and Erdős in 1976, several problems have been asked on estimating the minimum possible indepe…
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 positive integers $n\ge s$, we let ${\mathrm{ex}}(n,s)$ denote the maximum number of edges in a triangle-free graph $G$ on $n$ vertices with $α(G)\le s$. In the early sixties Andrásfai conjectured that for $n/3<s<n/2$ the function ${\mathrm{ex}}(n, s)$ is piecewise quadratic with critical values at $s/n={k}/({3k-1})$. We confirm that this is indeed the case whenever $s/n$ is slightly larger than a critical value, thus determining ${\mathrm{ex}}(n,s)$ for all $n$ and $s$ such that $s/n\in [{k}/({3k-1}), {k}/({3k-1})+γ_k]$, where $γ_k=Θ(k^{-6})$.
The discrete cube $\{0,1\}^d$ is a fundamental combinatorial structure. A subcube of $\{0,1\}^d$ is a subset of $2^k$ of its points formed by fixing $k$ coordinates and allowing the remaining $d-k$ to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no $r+1$ of them have non-empty intersection, how many pairwise intersections can …
A $ρ$-mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most $ρ$. For a graph $H$ and for $ρ\geq 1$, the {\em mean Ramsey-Turán number} $RT(n,H,ρ-mean)$ is the maximum number of edges a $ρ$-mean colored graph with $n$ vertices can have under the condition it does not have a monochromatic copy of $H$. It is conjectured that $RT(n,K_m,2-mean)=RT(n,K_m,2)$ where $RT(n,H,k)$ is the maximum number of edges a $k$ edge-colored graph with $n$ vertices can have under the condition it does not have a monochromatic copy of $H$. We p…
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 $f(n)$ be a function and $L$ be a graph. Denote by $RT(n,L,f(n))$ the maximum number of edges of an $L$-free graph on $n$ vertices with independence number less than $f(n)$. Erd\H os and Sós asked if $RT\left(n, K_5, c\sqrt{n}\right) = o(n^2)$ for some constant $c$. We answer this question by proving the stronger $RT\left(n, K_5, o\left(\sqrt{n\log n}\right)\right) = o(n^2)$. It is known that $RT \left(n, K_5, c \sqrt{n\log n} \right) = n^2/4+o(n^2)$ for $c>1$, so one can say that $K_5$ has a Ramsey-Turán phase transition at $c\sqrt{n\log n}$. We extend this result to several other $K_s$'s…
A seminal result of Hajnal and Szemerédi states that if a graph $G$ with $n$ vertices has minimum degree $δ(G) \ge (r-1)n/r$ for some integer $r \ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples which show optimality of the bound on $δ(G)$ are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Turán theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results: $\bullet$ For any $r > \ell \ge…
Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$, and $E_2=\{v_0v_1,\ldots,$ $v_0v_{n-3},v_{n-3}v_{n-2}, v_{n-3}v_{n-1}\}$. In this paper, for $p\ge n\ge 5$ we obtain explicit formulas for $\ex(p;T_n^1)$ and $\ex(p;T_n^2)$, where $\ex(p;L)$ denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph. Let $r(G\sb 1, G\sb 2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. In this paper we also obtain some explicit formulas for $r(T_m,…
One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graphs on $n$ vertices has at most $\lfloor n^2/4\rfloor$ edges. About half a century later Andrásfai studied dense triangle-free graphs and proved that the largest triangle-free graphs on $n$ vertices without independent sets of size $αn$, where $2/5\le α< 1/2$, are blow-ups of the pentagon. More than 50 further years have elapsed since Andrásfai's work. In this article we make the next step towards understanding the structure of dense triangle-free graphs without large independent sets. Notably,…
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…
We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{\text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Turán number $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$ $\bigg($ respectively, $\text{ex}(K_{n_{1},n_{2},n_{3}},\{C_{3}, C_{4}^{\text{multi}}\})$ $\bigg)$ is the maximum number of edges in a graph $G\subseteq K_{n_{1},n_{2},n_{3}}$ such that $G$ contains no $C_{4}^{\text{multi}}$ $\bigg($ respectively, $G$ contains neither $C_{3}$ nor $C_{4}^{\text{multi}}$ $\bigg)$. We call a $C^{multi}_4$ rainbow if a…
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…
We find the exact value of the Ramsey number $R(C_{2\ell},K_{1,n})$, when $\ell$ and $n=O(\ell^{10/9})$ are large. Our result is closely related to the behaviour of Turán number $ex(N, C_{2\ell})$ for an even cycle whose length grows quickly with $N$.
We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for $t \ge 3$, we show that any two-colouring of the complete graph on $n$ vertices that is $δ$-far from being monochromatic contains an \emph{unavoidable $t$-colouring} when $δ\gg n^{-1/t}$, where an unavoidable $t$-colouring is any two-colouring of a clique of order $2t$ in which one colour forms either a clique of order $t$ or two disjoint cliques of order $t$. Next, for $ t\ge 3$, we show that any tournament on $n$ vertices that is $δ$-far from being transitive contains an \emph{u…
In a seminal paper from 1983, Burr and Erdos started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving all but one of the Burr-Erdos problems.
We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.
HUN-REN Alfréd Rényi Institute of Mathematics（HUN-REN 阿尔弗雷德·雷尼数学研究所）；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）
We find the asymptotic value of the Ramsey number for a triple of long cycles, where the lengths of the cycles are large but may have different parity.
A Berge-$K_4$ in a triple system is a configuration with four vertices $v_1,v_2,v_3,v_4$ and six distinct triples $\{e_{ij}: 1\le i< j \le 4\}$ such that $\{v_i,v_j\}\subset e_{ij}$ for every $1\le i<j\le 4$. We denote by $\cal{B}$ the set of Berge-$K_4$ configurations. A triple system is $\cal{B}$-free if it does not contain any member of $\cal{B}$. We prove that the maximum number of triples in a $\cal{B}$-free triple system on $n\ge 6$ points is obtained by the balanced complete $3$-partite triple system: all triples $\{abc: a\in A, b\in B, c\in C\}$ where $A,B,C$ is a partition of $n$ poin…
A {\em special four-cycle } $F$ in a triple system consists of four triples {\em inducing } a $C_4$. This means that $F$ has four special vertices $v_1,v_2,v_3,v_4$ and four triples in the form $w_iv_iv_{i+1}$ (indices are understood $\pmod 4$) where the $w_j$s are not necessarily distinct but disjoint from $\{v_1,v_2,v_3,v_4\}$. There are seven non-isomorphic special four-cycles, their family is denoted by $\cal{F}$. Our main result implies that the Turán number $\text{ex}(n,{\cal{F}})=Θ(n^{3/2})$. In fact, we prove more, $\text{ex}(n,\{F_1,F_2,F_3\})=Θ(n^{3/2})$, where the $F_i$-s are specif…
The anti-Ramsey problem was introduced by Erdős, Simonovits and Sós in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s, \mathcal{H})$, is the smallest integer $c$ such that in any coloring of the edges of the $s$-uniform complete hypergraph on $n$ vertices with exactly $c$ colors, there is a copy of $\mathcal{H}$ whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large $n$, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtai…
Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán) properties of (hyper)graphs and the integers. In particular: (i) We generalise the random Ramsey …
