A result in asymmetric Euclidean Ramsey theory

It is proved that if the points of the three-dimensional Euclidean space are coloured with red and blue, then there exist either two red points at unit distance, or six collinear blue points with distance one between any two consecutive points.     

Transitive sets in Euclidean Ramsey theory

A finite set X in some Euclidean space Rn is called Ramsey if for any k there is a d such that whenever Rd is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erd?s, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory.In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever ‘rival’ conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively—in other words, if all points of the set look the same—our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement.Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest.     

Hypergraph coloring up to condensation

Improving a result of Dyer, Frieze and Greenhill [Journal of Combinatorial Theory, Series B, 2015], we determine the q‐colorability threshold in random k‐uniform hypergraphs up to an additive error of , where . The new lower bound on the threshold matches the “condensation phase transition” predicted by statistical physics considerations [Krzakala et al., PNAS 2007].     

Equitable list coloring and treewidth

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most ?|V(G)|/k? vertices. A graph is equitably k ‐choosable if such a coloring exists whenever the lists all have size k. Kostochka, Pelsmajer, and West introduced this notion and conjectured that G is equitably k‐choosable for k>Δ(G). We prove this for graphs of treewidth w≤5 if also k≥3w?1. We also show that if G has treewidth w≥5, then G is equitably k‐choosable for k≥max{Δ(G)+w?4, 3w?1}. As a corollary, if G is chordal, then G is equitably k‐choosable for k≥3Δ(G)?4 when Δ(G)>2. © 2009 Wiley Periodicals, Inc. J Graph Theory     

关于Reed列表染色猜想的一些结果

构造了一个图G,给G的每个顶点v一个颜色列表,使得每个列表Lv的大小至少为每个顶点v的邻域NG(v)与每个Vc交集的最大数目,但是这个图不存在一个正常的列表染色,从而推翻了R eed的一个猜想.     

On a list coloring conjecture of Reed

We construct graphs with lists of available colors for each vertex, such that the size of every list exceeds the maximum vertex‐color degree, but there exists no proper coloring from the lists. This disproves a conjecture of Reed. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 106–109, 2002     

List coloring triangle‐free hypergraphs

A triangle in a hypergraph is a collection of distinct vertices u, v, w and distinct edges e, f, g with , and . Johansson [Tech. report (1996)] proved that every triangle‐free graph with maximum degree Δ has list chromatic number . Frieze and Mubayi (Electron J Comb 15 (2008), 27) proved that every linear (meaning that every two edges share at most one vertex) triangle‐free triple system with maximum degree Δ has chromatic number . The restriction to linear triple systems was crucial to their proof. We provide a common generalization of both these results for rank 3 hypergraphs (edges have size 2 or 3). Our result removes the linear restriction from 8 , while reducing to the (best possible) result [Johansson, Tech. report (1996)] for graphs. In addition, our result provides a positive answer to a restricted version of a question of Ajtai Erd?s, Komlós, and Szemerédi (combinatorica 1 (1981), 313–317) concerning sparse 3‐uniform hypergraphs. As an application, we prove that if is the collection of 3‐uniform triangles, then the Ramsey number satisfies for some positive constants a and b. The upper bound makes progress towards the recent conjecture of Kostochka, Mubayi, and Verstraëte (J Comb Theory Ser A 120 (2013), 1491–1507) that where C₃ is the linear triangle. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 487–519, 2015     

List Ramsey numbers

We introduce a list‐coloring extension of classical Ramsey numbers. We investigate when the two Ramsey numbers are equal, and in general, how far apart they can be from each other. We find graph sequences where the two are equal and where they are far apart. For $?$‐uniform cliques we prove that the list Ramsey number is bounded by an exponential function, while it is well known that the Ramsey number is superexponential for uniformity at least 3. This is in great contrast to the graph case where we cannot even decide the question of equality for cliques.     

A list analogue of equitable coloring

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when unless G contains or k is odd and . For forests, the threshold improves to . If G is a 2-degenerate graph (given k ≥ 5) or a connected interval graph (other than ), then G is equitably k-choosable when . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 166–177, 2003     

On list coloring Steiner triple systems

We study the list chromatic number of Steiner triple systems. We show that for every integer s there exists n₀=n₀(s) such that every Steiner triple system on n points STS(n) with n≥n₀ has list chromatic number greater than s. We also show that the list chromatic number of a STS(n) is always within a log n factor of its chromatic number. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 314–322, 2009     

On Ramsey numbers of uniform hypergraphs with given maximum degree

For every ?>0 and every positive integers Δ and r, there exists C=C(?,Δ,r) such that the Ramsey number, R(H,H) of any r-uniform hypergraph H with maximum degree at most Δ is at most C|V(H)|^1+?.     

Acyclic list 7‐coloring of planar graphs

The acyclic list chromatic number of every planar graph is proved to be at most 7. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 83–90, 2002     

A note on list edge and list total coloring of planar graphs without adjacent short cycles

LetGbe a planar graph with maximum degreeΔ.In this paper,we prove that if any4-cycle is not adjacent to ani-cycle for anyi∈{3,4}in G,then the list edge chromatic numberχl(G)=Δand the list total chromatic numberχl(G)=Δ+1.     

On Generalized Ramsey Numbers for 3‐Uniform Hypergraphs

The well‐known Ramsey number is the smallest integer n such that every ‐free graph of order n contains an independent set of size u. In other words, it contains a subset of u vertices with no K₂. Erd?s and Rogers introduced a more general problem replacing K₂ by  for . Extending the problem of determining Ramsey numbers they defined the numbers where the minimum is taken over all ‐free graphs G of order n. In this note, we study an analogous function for 3‐uniform hypergraphs. In particular, we show that there are constants c₁ and c₂ depending only on s such that      

List edge and list total coloring of 1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that each 1-planar graph with maximum degree Δ is (Δ+1)-edge-choosable and (Δ+2)- total-choosable if Δ≥16, and is Δ-edge-choosable and (Δ+1)-total-choosable if Δ≥21. The second conclusion confirms the list coloring conjecture for the class of 1-planar graphs with large maximum degree.     

Equitable list coloring of planar graphs without 4- and 6-cycles

A graph G is equitably k-choosable if for any k-uniform list assignment L, there exists an L-colorable of G such that each color appears on at most vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k-choosable for k>Δ(G). We prove this for planar graphs with Δ(G)≥6 and no 4- or 6-cycles.     

On the Size‐Ramsey Number of Hypergraphs

The size‐Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2‐edge‐coloring of H yields a monochromatic copy of G. Size‐Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size‐Ramsey numbers for k‐uniform hypergraphs. Analogous to the graph case, we consider the size‐Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size‐Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.     

Maximum average degree and relaxed coloring

We say a graph is $(d,d,\dots ,d,0,\dots ,0)$-colorable with $a$ of $d$’s and $b$ of $0$’s if $V\left(G\right)$ may be partitioned into $b$ independent sets $O_1,O_2,\dots ,O_b$ and $a$ sets $D_1,D_2,\dots ,D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad\left(G\right)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a,b$, we show that if $mad\left(G\right)<\frac{4}{3}a+b$, then $G$ is $(1_1,1_2,\dots ,1_a,0_1,\dots ,0_b)$-colorable.