Rahil Baber, John Talbot
We prove a vertex domination conjecture of Erd\H os, Faudree, Gould, Gyárfás, Rousseau, and Schelp, that for every n-vertex complete graph with edges coloured using three colours there exists a set of at most three vertices which have at least 2n/3 neighbours in one of the colours. Our proof makes extensive use of the ideas presented in "A New Bound for the 2/3 Conjecture" by Král', Liu, Sereni, Whalen, and Yilma.
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Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erdős-Faudree-Schelp conjectures
本文证明了两个1981年的猜想，探讨了图对的Ramsey有限性，并反驳了Burr的更强猜想，同时重新表述了Faudree的定理。
Erdős, Faudree, Rousseau and Schelp observed the following fact for every fixed integer $k\geq 2$: Every graph on $n\geq k-1$ vertices with at least $(k-1)(n-k+2)+{k-2\choose 2}$ edges contains a subgraph with minimum degree at least $k$. However, there are examples in which the whole graph is the only such subgraph. Erdős et al. conjectured that having just one more edge implies the existence of a subgraph on at most $(1-\varepsilon_k)n$ vertices with minimum degree at least $k$, where $\varepsilon_k>0$ depends only on $k$. We prove this conjecture, using and extending ideas of Mousset, Noeve…
For positive integers $n>d\geq k$, let $ϕ(n,d,k)$ denote the least integer $ϕ$ such that every $n$-vertex graph with at least $ϕ$ vertices of degree at least $d$ contains a path on $k+1$ vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function $ϕ(n,d,k)$, and conjectured that for any positive integers $n>d\geq k$, it holds that $ϕ(n,d,k)\leq \lfloor\frac{k-1}{2}\rfloor\lfloor\frac{n}{d+1}\rfloor+ε$, where $ε=1$ if $k$ is odd and $ε=2$ otherwise. In this paper we determine the values of the function $ϕ(n,d,k)$ exactly. This confirms the above conjecture…
The Erdős-Faudree Problems and the Isolate-Free Core
本文研究了埃尔德什-福德里问题，证明了无孤立顶点核心决定了序列的行为，解决了问题1的正反答案及问题2的正向结论。
A spectral Erdős-Faudree-Rousseau theorem
本文利用超饱和-稳定性和谱技术证明了谱Erdős-Faudree-Rousseau定理，并展示了该方法在图书大小猜想、友谊图子结构及奇圈稳定性等问题中的广泛应用。
The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erdős, Faudree, Rousseau and Schelp conjectured that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ provided $\ell \geq C\frac {\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon >0$ and $n>…
Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ for which each vertex weight $f^V(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ is unique amongst all $v\in V(G)$. In 1987, Faudree and Lehel conjectured that there is a constant $c$ such that $s(G) \leq n/d + c$ for all $d$-regular graphs $G$ on $n$ vertices with $d>1$, whereas it is trivial that $s(G) \geq n/d$. In this short note we prove that the Faudree-Lehel Conjecture holds when $d \geq n^{0.8+ε}$ …
On a conjecture of Faudree and Schelp
该研究针对Faudree和Schelp关于哈密顿连通图路径长度的猜想，构造了三次平面反例补充Thomassen的结果，还构造了圈谱前半段有大量间隙的无穷哈密顿连通图族。
In this paper we prove several results on Ramsey numbers $R(H,F)$ for a fixed graph $H$ and a large graph $F$, in particular for $F = K_n$. These results extend earlier work of Erdős, Faudree, Rousseau and Schelp and of Balister, Schelp and Simonovits on so-called Ramsey size-linear graphs. Among others, we show that if $H$ is a subdivision of $K_4$ with at least $6$ vertices, then $R(H,F) = O(v(F) + e(F))$ for every graph $F$. We also conjecture that if $H$ is a connected graph with $e(H) - v(H) \leq \binom{k+1}{2} - 2$, then $R(H,K_n) = O(n^k)$. The case $k=2$ was proved by Erdős, Faudree, R…
The irregularity strength of a graph $G$, $s(G)$, is the least $k$ admitting a $\{1,2,\ldots,k\}$-weighting of the edges of $G$ assuring distinct weighted degrees of all vertices, or equivalently the least possible maximal edge multiplicity in an irregular multigraph obtained of $G$ via multiplying some of its edges. The most well-known open problem concerning this graph invariant is the conjecture posed in 1987 by Faudree and Lehel that there exists a constant $C$ such that $s(G)\leq \frac{n}{d}+C$ for each $d$-regular graph $G$ with $n$ vertices and $d\geq 2$ (while a straightforward countin…
Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted by $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ attributing distinct weighted degrees: $\tilde f(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ to all vertices $v\in V(G)$. It is straightforward that $s(G) \geq n/d$ for every $d$-regular graph $G$ on $n$ vertices with $d>1$. In 1987, Faudree and Lehel conjectured in turn that there is an absolute constant $c$ such that $s(G) \leq n/d + c$ for all such graphs. Even though the conjecture has remained o…
The Ramsey threshold for trees versus odd cycles
该研究解决了Burr等人提出的树与奇环的拉姆齐阈值问题，确定足够大奇数m对应的f(m)=⌈(2m−1)/3⌉，证实了Huang等人的相关猜想。
The Erdős-Sós Conjecture states that if $G$ is a simple graph of order $n$ with average degree more than $k-2,$ then $G$ contains every tree of order $k$. In this paper, we prove that Erdős-Sós Conjecture is true for $n=k+4$.
A famous conjecture of Erdős and Sós states that every graph with average degree more than $k - 1$ contains all trees with $k$ edges as subgraphs. We prove that the Erdős-Sós conjecture holds approximately, if the size of the embedded tree is linear in the size of the graph, and the maximum degree of the tree is sublinear.
The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs.
The Brown-Erdős-Sós conjecture, one of the central conjectures in extremal combinatorics, states that for any integer $m\geq 6,$ if a 3-uniform hypergraph on $n$ vertices contains no $m$ vertices spanning at least $m-3$ edges, then the number of edges is $o(n^2).$ We prove the conjecture for triple systems coming from finite abelian groups.
We prove the Erd\H os--Sós conjecture for trees with bounded maximum degree and large dense host graphs. As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a constant.
Let $G^n$ be a graph on $n$ vertices and let $w_k(G^n)$ denote the number of walks of length $k$ in $G^n$ divided by $n$. Erdős and Simonovits conjectured that $w_k(G^n)^t \geq w_t(G^n)^k$ when $k\geq t$ and both $t$ and $k$ are odd. We prove this conjecture.
The Erdös-Hajnal conjecture states that for every graph $H$, there exists a constant $δ(H) > 0$ such that every graph $G$ with no induced subgraph isomorphic to $H$ has either a clique or a stable set of size at least $|V(G)|^{δ(H)}$. This paper is a survey of some of the known results on this conjecture.
We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges. This result can be viewed as a special case of the degree version of a well-known conjecture of Erdős on hypergraph matchings. Improving the work of Bollobás, Daykin, and Erd\H os from 1976, we show that given integers $n, k, s$ with $n\ge 3k^2 s$, every $k$-uniform hypergraph $H$ on $n$ vertices with minimum vertex degree greater than $\binom{n-1}{k-1}-\binom…
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 $\mathrm{ex}(n,H)=O(n^{4/3+\varepsilon})$.
The Erdős-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability versions of this result, which confirm a prediction of Kahn.
We consider the Erd{ő}s - Faber - Lovász (EFL) conjecture for hypergraphs. This paper gives an upper bound for the chromatic number of $r$ regular linear hypergraphs $\textbf{H}$ of size $n$. If $r \ge 4$, $χ(\textbf{H}) \le 1.181n$ and if $r=3$, $χ(\textbf{H}) \le 1.281n$
We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value, proved by Erdos, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.
The Erdős-Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erdős, Tuza, and Valtr strengthened the conjecture by stating that any set of more than $\sum_{i = n - b}^{a - 2} \binom{n - 2}{i}$ points in a plane either contains the vertices of a convex $n$-gon, $a$ points lying on a concave downward curve, or $b$ points lying on a concave upward curve. They also showed that the generalization is actually equivalent to the Erdős-Szekeres conjecture. We prove the first new case of the Erdős-Tuza-Val…
We prove the well-known Brown-Erdős-Sós Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is large enough given the linear density of $G$, and the number of vertices of $G$ is large enough given $r$ and $k$.
The long-standing Erdős-Faber-Lovász conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős-Faber-Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations,…
A counterexample to the Erdős--Sós bipartite-link conjecture
该研究构造反例否定了Erdős与Sós提出的3-均匀超图相关猜想，反例边密度大于1/4，对足够大的n成立。
A proof of the Erdős-Gallai cycle decomposition conjecture
本文证明了 Erdős--Gallai 猜想：任意 n 顶点图的边可分解为 O(n) 个环和边，并由此推出 Hajós 猜想在常数因子意义下成立。
A graph is $P_8$-free if it contains no induced subgraph isomorphic to the path $P_8$ on eight vertices. In 1995, Erdős and Gyárfás conjectured that every graph of minimum degree at least three contains a cycle whose length is a power of two. In this paper, we confirm the conjecture for $P_8$-free graphs by showing that there exists a cycle of length four or eight in every $P_8$-free graph with minimum degree at least three.
An old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$-vertex graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, Ω(n^2)\}$. We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, Ω(n^2)\}$.
Alon, Frankl, and Lovász proved a conjecture of Erdős that one needs at least $\lceil \frac{n-r(k-1)}{r-1} \rceil$ colors to color the $k$-subsets of $\{1, \dots, n\}$ such that any $r$ of the $k$-subsets that have the same color are not pairwise disjoint. A generalization of this problem where one requires $s$-wise instead of pairwise intersections was considered by Sarkaria. He claimed a proof of a generalized Erdős--Kneser conjecture establishing a lower bound for the number of colors that reduces to Erdős' original conjecture for ${s = 2}$. Lange and Ziegler pointed out that his proof fail…
On monochromatic path covers conjecture of Erdős--Gyárfás
研究埃尔德什 - 贾尔法什关于完全图\(K_n\)红 - 蓝边着色的单色路径覆盖猜想，采用极小反例法，完全证实了该猜想，此前已有对足够大\(n\)的证明。
The Brown-Erdős-Sós conjecture in dense triple systems
本文证明了著名的Brown-Erdős-Sós猜想对于稠密线性3一致超图在δ>4/5时成立，确立了首个此类界限。
We prove that for every graph $H$, there exists $\varepsilon>0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $|A|, |B|\ge \varepsilon n$ and $A$ is complete or anticomplete to $B$. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erdős-Hajnal conjecture for vertex-minors.
Bounded VC-dimension implies the Erdős--Rado sunflower conjecture
本文证明了对于有界VC维的族，Erdős--Rado向日葵猜想成立，即存在常数$K(r)$使得成员数超过$K(r)^{\ell}$的$\ell$-均匀族必含$r$-向日葵。
A linear hypergraph is intersecting if any two different edges have exactly one common vertex and an $n$-quasicluster is an intersecting linear hypergraph with $n$ edges each one containing at most $n$ vertices and every vertex is contained in at least two edges. The Erdös-Faber-Lovász Conjecture states that the chromatic number of any $n$-quasicluster is at most $n$. In the present note we prove the correctness of the conjecture for a new infinite class of $n$-quasiclusters using a specific edge coloring of the complete graph.
In this paper, we prove an enhanced version of the Erdős-Lovász Tihany Conjecture for line graphs of multigraphs. That is, for every graph $G$ whose chromatic number $χ(G)$ is more than its clique number $ω(G)$ and for nonnegative integer $\ell$, any two integers $s,t \geq 3.5\ell+2$ with $s+t = χ(G)+1$, there is a partition $(S,T)$ of the vertex set $V(G)$ such that $χ(G[S])\geq s$ and $χ(G[T])\geq t+\ell$. In particular, when $\ell=1$, we can obtain the same result just for any $s,t\geq4$. The Erdős-Lovász Tihany conjecture is a special case when $\ell=0$.
In 1965 Erd\H os conjectured that for all $k\ge2$, $s\ge1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\F$ with $ν(\F)=s$ cannot have more than \newline $\max\{\binom{sk+k-1}k,\;\binom nk-\binom{n-s}k\}$ edges. It took almost fifty years to prove it for triple systems. In 2012 we proved the conjecture for all $s$ and all $n\ge4(s+1)$. Then Łuczak and Mieczkowska (2013) proved the conjecture for sufficiently large $s$ and all $n$. Soon after, Frankl proved it for all $s$. Here we present a simpler version of that proof which yields Erd\H os's conjecture for $s\ge33$. Our motivation…
In 1968, Erdös and Lovász conjectured that for every graph $G$ and all integers $s,t\geq 2$ such that $s+t-1=χ(G) > ω(G)$, there exists a partition $(S,T)$ of the vertex set of $G$ such that $χ(G|S)\geq s$ and $χ(G|T)\geq t$. For general graphs, the only settled cases of the conjecture are when $s$ and $t$ are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2 \cite{quasi-line}, line graphs \cite{line} and quasi-line graphs \cite{quasi-line}. In this paper, we consider the conjecture for claw-free graphs and present some progress on i…
The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the central problems in combinatorics. One of the oldest results in Ramsey Theory, proved by Erdős and Szekeres in 1935, asserts that the Ramsey number of the complete graph with $m$ edges is at most $2^{O(\sqrt{m})}$. Motivated by this estimate Erdős conjectured, more than a quarter century ago, that there is an absolute constant $c$ such that $r(G) \leq 2^{c\sqrt{…
The Erd\H os-Sós conjecture in dense graphs
该研究证明了稠密图中的Erdős–Sós猜想，即当n足够大且k不小于γn时，边数超标的n顶点图包含所有k顶点树，还解决了Erdős与Graham关于树的多色拉姆齐数的51年难题。
A well-known conjecture of Erdős and Sós states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding $m$ and minimum degree at least $\lfloor \frac{2m}{3}\rfloor$ contains every tree with $m$ edges. As evidence for our conjecture we show (i) for every $m$ there is a $g(m)$ such that the weakening of the conjecture obtained by replacing $m$ by $g(m)$ holds, and (ii) there is a $γ>0$ such that the weakening of the conjecture obtained by replacing $\…
Let $s\ge2$ and $t\ge2$ be integers. A graph $G$ is $(s,t)$-\emph{splittable} if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $χ(G[S])\geq s$ and $χ(G[T])\geq t$. The well-known Erdős-Lovász Tihany Conjecture from 1968 states that every graph $G$ whose chromatic number $χ(G)=s+t-1$ is more than its clique number $ω(G)$ is $(s,t)$-splittable. In this paper, we prove an enhanced version of the Erdős-Lovász Tihany Conjecture for graphs with independence number two. That is, for every graph $G$ with $χ(G)=s+t-1>ω(G)+1$ is $(s,t+1)$-splittable. There are examples showing that this …
The asymptotic Erd{ő}s--Frankl--Füredi cover-free conjecture
该研究证实了1985年Erdős等人提出的渐近Erdős–Frankl–Füredi无覆盖猜想，证明了r-无覆盖族的阈值T(r)满足T(r)≥r²+r+2（r≥1）。
Let $Q_n$ be the $n$-dimensional hypercube, and let ${\rm cr}(Q_n)$ be the \textit{crossing number} of $Q_n$. Erdős and Guy in 1973 conjectured the following equality: ${\rm cr}(Q_n)=\frac{5}{32}4^n-\lfloor\frac{n^2+1}{2}\rfloor 2^{n-2}$. In this paper, we construct a drawing of $Q_n$ with less crossings when $n>6$, which implies that for $n>6$ we have a strict inequality.
An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erdős, Faudree and Sós conjectured that in any C-Ramsey graph there are $Ω\left(n^{5/2}\right)$ induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samo…
The Erdős-Sós Conjecture states that every graph with average degree more than $k-2$ contains all trees of order $k$ as subgraphs. In this paper, we consider a variation of the above conjecture: studying the maximum size of an $(n,m)$-bipartite graph which does not contain all $(k,l)$-bipartite trees for given integers $n\ge m$ and $k\ge l$. In particular, we determine that the maximum size of an $(n,m)$-bipartite graph which does not contain all $(n,m)$-bipartite trees as subgraphs (or all $(k,2)$-bipartite trees as subgraphs, respectively). Furthermore, all these extremal graphs are characte…
The Erdős--Chvátal Simplex Conjecture
证明 Erdős--Chvátal 单纯形猜想：在给定参数条件下，d-单纯形-free 的 k-子集族大小不超过二项式系数，等号仅全星。研究问题为极值组合中的单纯形猜想，方法为证明整个参数范围，贡献为完整解决该猜想。
We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order logic formulae concerning some sets of multisets of uniform cardinality over boolean variables would prove the Erdös-Szekeres conjecture, which states that for any set of 2^(n-2)+1 points in general position, there exists n points forming a convex polygon, where n is greater than or equal to 3.
Three graphs related to the \EGC\, are presented. The graphs are derived from the Buckyball, the Petersen graph, and the Tutte-Coxeter graph. The first graph is a partial answer to a question posed by Heckman and Krakovski \cite{planar} in their recent work on the planar version of the conjecture. The other two graphs appear to be the smallest known cubic graphs with no $2^m$-cycles for $m \leq 4$ and for $m \leq 5$.
A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erdős conjectured that one can find so-called `sparse' Steiner triple systems. Roughly speaking, the aim is to have at most $j-3$ triples on every set of $j$ points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural generalization of the triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and Rödl as…
We prove a new upper bound for the minimum $d$-degree threshold for perfect matchings in $k$-uniform hypergraphs when $d<k/2$. As a consequence, this determines exact values of the threshold when $0.42k \le d < k/2$ or when $(k,d)=(12,5)$ or $(17,7)$. Our approach is to give an upper bound on the Erdős Matching Conjecture and convert the result to the minimum $d$-degree setting by an approach of Kühn, Osthus and Townsend. To obtain exact thresholds, we also apply a result of Treglown and Zhao.
Erdös's Matching Conjecture and $s$-wise $t$-intersection Conjecture via Symmetrical Smoothing Method
本研究借助经小幅修正的对称平滑法，推导出不含ℓ匹配的n元组族最大基数公式（可简化为Erdös匹配公式），并证明了k元子集的最大s-wise t-相交族基数猜想。
The Erdős--Faber--Lovász Conjecture, posed in 1972, states that if a graph $G$ is the union of $n$ cliques of order $n$ (referred to as defining $n$-cliques) such that two cliques can share at most one vertex, then the vertices of $G$ can be properly coloured using $n$ colours. Although still open after almost 50 years, it can be easily shown that the conjecture is true when every shared vertex belongs to exactly two defining $n$-cliques. We here provide a quick and easy algorithm to colour the vertices of $G$ in this case, and discuss connections with clique-decompositions and edge-colourings…
Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures
本文证明了多项式Rödl猜想、多项式Nikiforov猜想与Erdős-Hajnal猜想三者等价，并由此推出字符串图满足多项式Rödl猜想，同时将结果推广至超图、锦标赛、有序图和着色图。
In 1965, Paul Erdős asked about the largest family $Y$ of $k$-sets in $\{ 1, \ldots, n \}$ such that $Y$ does not contain $s+1$ pairwise disjoint sets. This problem is commonly known as the Erdős Matching Conjecture. We investigate the $q$-analog of this question, that is we want to determine the size of a largest family $Y$ of $k$-spaces in $\mathbb{F}_q^n$ such that $Y$ does not contain $s+1$ pairwise disjoint $k$-spaces. Here we call two subspaces disjoint if they intersect trivially. Our main result is, slightly simplified, that if $16 s \leq \min\{ q^{\frac{n-k}{4}},$ $q^{\frac{n-2k+1}{3}…
The conjecture of Brown, Erdős and Sós from 1973 states that, for any $k \ge 3$, if a $3$-uniform hypergraph $H$ with $n$ vertices does not contain a set of $k+3$ vertices spanning at least $k$ edges then it has $o(n^2)$ edges. The case $k=3$ of this conjecture is the celebrated $(6,3)$-theorem of Ruzsa and Szemerédi which implies Roth's theorem on $3$-term arithmetic progressions in dense sets of integers. Solymosi observed that, in order to prove the conjecture, one can assume that $H$ consists of triples $(a, b, ab)$ of some finite quasigroup $Γ$. Since this problem remains open for all $k …
The celebrated Erdös-Hajnal conjecture states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. This conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H$-free $n$-vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. This conjecture is proved for few infinite families of tournaments. In this paper we construct…
Asymptotically optimal Ramsey goodness of sparse graphs versus odd cycles and paths
本文研究稀疏图与奇数环和路径的渐近最优Ramsey良性，证明了对于奇数环n=Ω(k)足够，路径n=Ω(k^2)或附加条件n=Ω(k)。
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