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On generalized Kneser hypergraph colorings

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On generalized Kneser hypergraph colorings Carsten https://www.wendangku.net/doc/b515583331.html,nge ?University of Washington Department of Mathematics Box 354350Seattle,WA 98195-4350lange@math.tu-berlin.de G ¨u nter M.Ziegler ??TU Berlin Inst.Mathematics,MA 6-2D-10623Berlin,Germany ziegler@math.tu-berlin.de February 2006Abstract In Ziegler (2002),the second author presented a lower bound for the chromatic num-bers of hypergraphs KG r s S ,“generalized r -uniform Kneser hypergraphs with inter-section multiplicities s .”It generalized previous lower bounds by Kˇr ′?ˇz (1992/2000)for the case s =(1,...,1)without intersection multiplicities,and by Sarkaria (1990)for S = [n ]k .Here we discuss subtleties and di?culties that arise for intersection multiplicities s i >1:1.In the presence of intersection multiplicities,there are two di?erent versions of a “Kneser hypergraph,”depending on whether one admits hypergraph edges that are multisets rather than sets.We show that the chromatic numbers are substantially di?erent for the two concepts of hypergraphs.The lower bounds of Sarkaria (1990)and Ziegler (2002)apply only to the multiset version.2.The reductions to the case of prime r in the proofs by Sarkaria and by Ziegler work only if the intersection multiplicities are strictly smaller than the largest prime factor of r .Currently we have no valid proof for the lower bound result in the other cases.We also show that all uniform hypergraphs without multiset edges can be represented as generalized Kneser hypergraphs.

1Introduction

The “generalized Kneser hypergraphs with intersection multiplicities,”as studied in [10],arose from the graphs (implicitly)studied by Kneser [4]in several subsequent generalization

steps.They do form a large class of hypergraphs—indeed,we will show in Section3that every uniform hypergraph without multiplicities can be represented in this model.

When writing[10],the second author overlooked that the edges of the generalized Kneser hypergraph KG r s S with intersection multiplicities s=(s1,...,s n)could be multisets if s i>1:Their edges can have repeated elements,as pointed out in[7].Thus KG r s S is not a hypergraph in the traditional sense of Berge[2],where the edges have to be sets.If one does not allow for repeated elements in the edges,then this yields a sub-hypergraph without multiplicities,

kg r s(S)?KG r s(S).

Both for kg r s(S)and for KG r s(S)we are faced with the problem to determine the chromatic number:How many colors are needed for the sets in S if monochromatic hypergraph edges are forbidden?Clearly we haveχ(kg r s S)≤χ(KG r s S),but the two values can be far apart, as we will see below.

In this note,we discuss the chromatic numbers of generalized Kneser hypergraphs with intersection multiplicities,in view of some main topics and results from[10].This includes errata and clari?cations announced in[11]:

[10,Lemma3.1]described an explicit coloring for the special case of S= [n]k and constant s=(s,...,s).This coloring is valid for generalized Kneser hypergraphs with multiplicies,which yields

χ(kg r

s n k )≤χ(KG r s n k )≤1+ 1s?ns?rk+1

cd r s S?≤χ(KG r s S)

r?1

for the generalized Kneser hypergraphs.This lower bound holds for generalized Kneser hypergraphs with multiplicities and is not valid forχ(kg r s S)as we will see in Section4.

Moreover,Karsten Vogel(Magdeburg)has noticed that the reduction to the prime case in the proof of[10,Theorem5.1]fails when the intersection multiplicities are not smaller than the prime factors of r.As we will analyze in Section5,we get only the following theorem(with a combinatorial proof):

Theorem1.1.Let S?2[n]be a set family,and let the intersection multiplicities s i≥1be smaller than the largest prime factor of r≥2.Then

S)≥?1

χ(KG r

s

We still believe that the theorem is valid for arbitrary s i

Thm.5.1],but we have no proof for this generality—not even for the“complete uniform”case when S= [n]k ;indeed,we will see in Section5that the induction proof by Sarkaria[9]for this case is not valid,either.

[10,Example7.2]analyzed some Kneser hypergraphs without multiplicities,including a computation ofχ(kg4(2,...,2) [n]2 ).Thus in Section4we discuss families of hypergraphs that include KG4(2,...,2) [n]2 .We also collect evidence towards the conjecture that the upper boundχ(KG r(s,...,s) n k )≤1+ 1s?ns?rk+1

n.

The s-disjoint r-colorability defect cd r s S of a set S?2[n]is the minimal number of elements one has to remove from the multiset[n]s such that the remaining multiset can be covered by r subsets of[n]that do not contain any element from S:

cd r s S=

An r-uniform hypergraph in the sense of Berge[2](without multiplicities)is a pair(V,E) that consists of a?nite set of vertices V and a set E of edges,which are r-subsets of V.

An r-uniform hypergraph with multiplicities is a pair(V,E)that consists of a?nite set of vertices V and a set E of edges,which are r-multisubsets of V.

A hypergraph(with multiplicities)is loop-free if every edge has at least two distinct elements(vertices).In the following,all hypergraphs are supposed to be loop-free.

According to this de?nition,hypergraphs do not have multiple edges;this makes sense for our purposes,since multiple edges are irrelevant for coloring.

The loop-free r-uniform hypergraph analog of the complete graph K n on n vertices is denoted by K r n.The vertex set of K r n is[n],and the edges are all the r-multisubsets of[n] that contain at least two distinct vertices.Thus K r n has n r ?n= n+r?1r ?n edges.The analogous complete r-uniform hypergraph k r n without multiplicities has n r edges. De?nition2.4(r-uniform s-disjoint Kneser hypergraphs).For any?nite set S= {S1,...,S m}of non-empty subsets of[n],the r-uniform s-disjoint Kneser hypergraph KG r s S with multiplicities has the vertex set S and the edge set

E(KG r

s

S):={all s-disjoint r-multisets whose elements are sets S i∈S}.

If s i

The r-uniform s-disjoint Kneser hypergraph kg r s S without multiplicities has the same vertex set S,but all of its edges are sets rather than multisets:

E(kg r

s

S):={all s-disjoint r-subsets of S}.

The generalized Kneser hypergraphs kg r s S are loop-free for any s.

We use KG r s S as a shorthand for KG r(s,...,s)S in the case of constant intersection multi-plicity s=(s,...,s),and similarly we write kg r s S and cd r s S.

The previously de?ned complete r-uniform hypergraphs K r n and k r n are examples of

r-uniform s-disjoint Kneser hypergraphs.We have K r

n

=KG r r?1 [n]1 ,k r n=kg r r?1 [n]1 ,and in this particular situation KG r1 [n]1 =kg r r?1 [n]1 .

We can obtain kg r s S from KG r s S by discarding edges.In this sense,kg r s S is a subhy-pergraph of KG r s S.In the special case that s i≡1we have KG r s S=kg r s S since pairwise disjoint non-empty sets are distinct.In particular,for r=2and s i≡1both de?nitions specialize to the generalized Kneser graph of S?2[n].

De?nition2.5(hypergraph colorings[3]).A coloring of an r-uniform hypergraph H (multiplicity-free or not)with m colors is a map c:V(H)→[m]that assigns to each vertex of H a color such that no edge is monochromatic,that is,for each e∈E(H)we have|{c(x)|x∈e}|≥2.Any coloring c of H by m colors induces a homomorphism

H→K r

m of hypergraphs.The chromatic numberχ(H)is the smallest number m such

that there is a coloring of H with m colors.

4

3How general are generalized Kneser hypergraphs?Matouˇs ek &Ziegler [8,p.76]observed that every (?nite,simple)graph can be represented as a Kneser graph:For any G =(V,E )there is a set system S ={S v |v ∈V }?2[m ],for some m ,such that S v ∩S w =?if and only if {v,w }∈E .Thus it is natural to ask which hypergraphs (with or without multiplicities)can be represented as generalized Kneser hypergraphs.The following proposition collects our answers to this question.For this,call a hypergraph up-monotone if for e,e ′∈ [n ]r with e ∈E we also have e ′∈E whenever the support of e ′contains that of e .Every r -uniform hypergraph without multiplicities is up-monotone,as is every generalized Kneser hypergraph KG r r ?1S .A hypergraph H =([n ],E )is convex if every integral weight vector (a 1,...,a n )in the convex hull of multiplicity vectors of edges of H (thus 0≤a i

Proposition 3.1.

(1)There are r -uniform hypergraphs without multiplicities that cannot be represented as a Kneser hypergraph KG r 1S .

(2)An r -uniform hypergraph H =([n ],E )with multiplicities can be represented as KG r r ?1S if and only if it is up-monotone.

In particular,every r -uniform hypergraph without multiplicities can be represented as a Kneser hypergraph KG r r ?1S .

(3)If an r -uniform hypergraph is representable by a generalized Kneser hypergraph with intersection multiplicities then it is convex.(The converse is not true.)

In particular,there are r -uniform hypergraphs with multiplicities that cannot be repre-sented as a Kneser hypergraph KG r s S .

Proof.(1).Consider ([4],{124,134,234}).If KG 31{S 1,...,S 4}has {S 1,S 2,S 4},{S 1,S 3,S 4}and {S 2,S 3,S 4}as edges,then each of the triples of sets is pairwise disjoint,so in particular S 1,S 2,S 3are pairwise disjoint.Thus also {S 1,S 2,S 3}is an edge in the Kneser hypergraph.

(2).The following construction generalizes the construction for graphs in [8].Let H =([n ],E )be up-monotone,and let ˉH

=([n ],ˉE )be the complementary hypergraph of H ,i.e.the hypergraph that has the same vertices as H and all edges of K r n that are not edges

of H .De?ne the set system S ={S i |i ∈[n ]}by

S i :={i }∪ ˉe ∈ˉE i ∈ˉe .

The S i are clearly distinct.If e ={{i 1,...,i r }}is an edge of H ,then

S i 1∩···∩S i r ={i 1}∩···∩{i r }∩ ˉe

∈ˉE i 1,...,i r ∈ˉe ,where the ?rst part is empty since H does not have loops (so the i k cannot all be equal)and the last set is empty since H is up-monotone.Thus S i 1∩···∩S i r is an edge of the

Kneser hypergraph.Conversely,if S i 1∩···∩S i r =?,then in particular it does not contain

the element e ={i 1...,i r },so e ∈E .

5

(3).The intersection multiplicities s i de?ne the hypergraph KG r s S as a subgraph of K r m, for m=|S|,by linear conditions on the multiplicity vectors of the edges.

For an example consider([3],{113,223}).If KG3s{S1,S2,S3},with S1,S2,S3?[m],does not have{S1,S2,S3}as an edge,then there is some i∈[m]such that S1,S2,S3contain i more than s i times.However,that cannot be if both{{S1,S1,S3}}and{{S2,S2,S3}}are edges,so S1,S1,S3and S2,S2,S3contain i at most s i times.

An example of a convex uniform hypergraph with multiplicities that cannot be repre-sented as a generalized Kneser hypergraph is([3],{112,223}).

For the purpose of coloring,any hypergraph H=(V,E)with multiplicities can be re-placed by an up-monotone uniform hypergraph with multiplicities,on the same ground set, and with the same chromatic number:For this replace each edge e by all multisets of cardi-nality r which contain the support of e,for some large enough r.By Proposition3.1(2),the resulting r-uniform hypergraph with multiplicities H′can be represented as a generalized Kneser graph,which yields topological lower bounds forχ(H)in terms of the colorability defect of H′.In particular,this applies to(non-uniform)hypergraphs in the sense of Berge. 4Two counterexamples

The purpose of this section will be to show that the lower boundχ(KG r s S)≥?1

5The induction to non-prime cases

For the case of Theorem1.1when p is a prime Ziegler[10]has given a combinatorial proof; an alternative topological proof was given by Lange[7,Sect.4.4].The special case when S= [n]k is due to Sarkaria[9].

In this section,we show that the reductions of the situation with general r to the case of prime r by Sarkaria[9]and by Ziegler[10]are both incomplete.We also argue that argument given in[10]su?ces to establish the result in the generality of Theorem1.1. Sarkaria’s proof[9,(3.2)]starts with the assumption that KG p j?1 [N]S has an M-coloring, with(p?1)M

N′:=M(p2?1)+p2(S?1)+1,

and tries to get a monochromatic(j?1)-disjoint p1-tuple of N′-subsets of[N].The argument fails if N′is larger than N,so there won’t be any N′-subsets of[N].Concrete parameters where this happens are p=4,p1=p2=2,j=4,S=2,M=N?2,which yields N′=M+3=N+1.(The problem does not occur for j≤3,so in particular the proof specializes correctly to the case j=2treated by Alon,Frankl&Lov′a sz[1].) Ziegler’s reduction to the prime case in[10,pp.679-680]is an extension of Kˇr′?ˇz’proof[6], which in turn generalizes the argument of Alon,Frankl and Lov′a sz.Let r=r′r′′with r′≤r′′.The goal is to derive a contradiction if we assume that cd r s S>(r?1)χ(KG r s S).A crucial ingredient is the set

T:= N?[n] cd r′1S|N>(r′?1)χ(KG r s S)

where S|N denotes the elements of S that are subsets of N.One then wants to argue that

(r′′?1)χ(KG r′′s T)≥cd r′′s T

But this can be concluded by induction only if sr′′for some i and T=?:In this situationχ(KG r′′s?)=0since there are no vertices to color, but cd r′′s?>0since at least s i?r′′elements have to be removed removed from[n]s to cover the remaining elements with r′′subsets of[n]).The case T=?can occur,as we have seen above for the special case of S= [N]S .

Thus,[10,Thm.5.1]can currently only be established in the generality given above as Theorem1.1.To establish this,one uses the induction given at[10,pp.679-680],factoring non-prime r=r′r′′so that r′′is the largest prime number that divides r.

6More Examples

In[10,Sect.7]the second author had raised the question whether the upper bound of[10, Lemma3.1]

χ(KG r

) [n]k ≤1+ 1s?ns?rk+1

s

is always tight,for n≥k≥2,r>s≥2,rk≤sn.In[10,Example7.2]he had claimed that (?)is not sharp for KG42 [n]2 .However,this is not true:The analysis given there referred to the corresponding Kneser hypergraph without multiplicities,that is,it established that

χ(kg42 [n]2 )=n? 4?1

s

?=1,i.e.r

s =1+n? 2r?12?,an

edge of H cannot contain two disjoint edges from K n.Thus the edges of H are supported only on stars or on triangles—the latter is permitted if s≥2r

3.In

either case the greedy colorings that provide the upper bound are optimal:n?1colors are needed for2r

2

≤s<2r

r?1 n2 ?n

n??r

r?1 r2 ?=?r

r?1 n2 ?sets of size at most r?1.

In summary,we see that

n??r

2n+1

2 =χ(kg42 [n]2 )?χ(KG42 [n]2 )=n?1

for su?ciently large n and r.This shows a huge di?erence between the chromatic numbers of generalized Kneser hypergraphs with and without multiplicities.

References

[1]N.Alon,P.Frankl,and L.Lov′a sz,The chromatic number of Kneser hypergraphs,Trans-

actions Amer.Math.Soc.,298(1986),pp.359–370.

8

[2]C.Berge,https://www.wendangku.net/doc/b515583331.html,binatorics of Finite Sets,no.45in North-Holland Mathematical

Library,North-Holland,Amsterdam,1989.

[3]P.Erd?o s and A.Hajnal,On chromatic number of graphs and set-systems,Acta Math.

Acad.Sci.Hungar.,17(1966),pp.61–99.

[4]M.Kneser,Aufgabe360,Jahresbericht der Deutschen Mathematiker-Vereinigung,2.Abt.,

58(1955),p.27.

[5]I.Kriz,Equivariant cohomology and lower bounds for chromatic numbers,Transactions Amer.

Math.Soc.,33(1992),pp.567–577.

[6]I.Kriz,A correction to“Equivariant cohomology and lower bounds for chromatic numbers”,

Transactions Amer.Math.Soc.,352(2000),pp.1951–1952.

[7]https://www.wendangku.net/doc/b515583331.html,nge,Combinatorial Curvatures,Group Actions,and Colour-

ings:Aspects of Topological Combinatorics,PhD thesis,TU Berlin, 2004.http://edocs.tu-berlin.de/diss/2004/lange_carsten.htm;see also arXiv:math.CO/0312067.

[8]J.Matouˇs ek and G.M.Ziegler,Topological lower bounds for the chromatic number:A hi-

erarchy,Jahresbericht der DMV,106(2004),pp.71–https://www.wendangku.net/doc/b515583331.html,/math.CO/0208072.

[9]K.S.Sarkaria,A generalized Kneser conjecture,https://www.wendangku.net/doc/b515583331.html,binatorial Theory,Ser.B,49(1990),

pp.236–240.

[10]G.M.Ziegler,Generalized Kneser coloring theorems with combinatorial proofs,Inventiones

math.,147(2002),pp.671–https://www.wendangku.net/doc/b515583331.html,/math.CO/0103146.

[11]G.M.Ziegler,Erratum:Generalized Kneser coloring theorems with combinatorial proofs,

Inventiones math.,163(2006),pp.227-228.

9

in on at的时间用法和地点用法 完全版

in,on,at的时间用法和地点用法 一、in, on, at的时间用法 ①固定短语: in the morning/afternoon/evening在早晨/下午/傍晚, at noon/night在中午/夜晚, (不强调范围,强调的话用during the night) early in the morning=in the early morning在大清早, late at night在深夜 on the weekend在周末(英式用at the weekend在周末,at weekends每逢周末) on weekdays/weekends在工作日/周末, on school days/nights在上学日/上学的当天晚上, ②不加介词 this, that, last, next, every, one, yesterday, today, tomorrow, tonight,all,most等之前一般不加介词。如, this morning 今天早晨 (on)that day在那天(that day更常用些) last week上周 next year明年 the next month第二个月(以过去为起点的第二个月,next month以现在为起点的下个月) every day每天 one morning一天早晨 yesterday afternoon昨天下午 tomorrow morning明天早晨 all day/morning/night整天/整个早晨/整晚(等于the whole day/morning/night) most of the time (在)大多数时间 ③一般规则 除了前两点特殊用法之外,其他≤一天,用on,>一天用in,在具体时刻或在某时用at(不强调时间范围) 关于on 生日、on my ninth birthday在我九岁生日那天 节日、on Teachers’Day在教师节 (注意:节日里有表人的词汇先复数再加s’所有格,如on Children’s Day, on Women’s Day, on Teachers Day有四个节日强调单数之意思,on Mother’s Day, on Father’s Day, on April Fool’s Day, on Valenti Day) 星期、on Sunday在周日,on Sunday morning在周日早晨 on the last Friday of each month 在每个月的最后一个星期五 日期、on June 2nd在六月二日 on the second (of June 2nd) 在六月的第二天即在六月二日 on the morning of June 2nd在六月二日的早晨,on a rainy morning在一个多雨的早晨 on a certain day 在某天 on the second day在第二天(以过去某天为参照) 注意:on Sunday在周日,on Sundays每逢周日(用复数表每逢之意),every Sunday每个周日,基本一个意思。 on a school day 在某个上学日,on school days每逢上学日。on the weekend在周末,on weekends每逢 周末。 关于in in June在六月 in June, 2010在2010年六月

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分,有括号的作用。知识的问题是一个科学问题,来不得半点的虚伪和骄 傲,决定地需要的倒是其反面——诚实和谦逊的态度。2.表示意思的递进。 团结——批评和自我批评——团结3.表示意思的转折。很白很亮的一堆洋 钱!而且是他的——现在不见了!连接号⑥—1.表示时间、地点、数目等 的起止。抗日战争时期(1937-1945年)“北京—上海”直达快车2.表 示相关的人或事物的联系。亚洲—太平洋地区书名号⑦《》〈〉表示 书籍、文件、报刊、文章等的名称。《矛盾论》《中华人民共和国宪法》《人 民日报》《红旗》杂志《学习〈为人民服务〉》间隔号·1.表示月份和日期 之间的分界。一二·九运动2.表示某些民族人名中的音界。诺尔曼·白求 恩着重号.表示文中需要强调的部分。学习马克思列宁主义,要按照毛泽 东同志倡导的方法,理论联系实际。······

In on at 时间用法及练习

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常用标点符号用法含义

一、基本定义 句子,前后都有停顿,并带有一定的句调,表示相对完整的意义。句子前后或中间的停顿,在口头语言中,表现出来就是时间间隔,在书面语言中,就用标点符号来表示。一般来说,汉语中的句子分以下几种: 陈述句: 用来说明事实的句子。 祈使句: 用来要求听话人做某件事情的句子。 疑问句: 用来提出问题的句子。 感叹句: 用来抒发某种强烈感情的句子。 复句、分句: 意思上有密切联系的小句子组织在一起构成一个大句子。这样的大句子叫复句,复句中的每个小句子叫分句。 构成句子的语言单位是词语,即词和短语(词组)。词即最小的能独立运用的语言单位。短语,即由两个或两个以上的词按一定的语法规则组成的表达一定意义的语言单位,也叫词组。 标点符号是书面语言的有机组成部分,是书面语言不可缺少的辅助工具。它帮助人们确切地表达思想感情和理解书面语言。 二、用法简表 名称

句号① 问号符号用法说明。?1.用于陈述句的末尾。 2.用于语气舒缓的祈使句末尾。 1.用于疑问句的末尾。 2.用于反问句的末尾。 1.用于感叹句的末尾。 叹号! 2.用于语气强烈的祈使句末尾。 3.用于语气强烈的反问句末尾。举例 xx是xx的首都。 请您稍等一下。 他叫什么名字? 难道你不了解我吗?为祖国的繁荣昌盛而奋斗!停止射击! 我哪里比得上他呀! 1.句子内部主语与谓语之间如需停顿,用逗号。我们看得见的星星,绝大多数是恒星。 2.句子内部动词与宾语之间如需停顿,用逗号。应该看到,科学需要一个人贡献出毕生的精力。 3.句子内部状语后边如需停顿,用逗号。对于这个城市,他并不陌生。 4.复句内各分句之间的停顿,除了有时要用分号据说苏州园林有一百多处,我到过的不外,都要用逗号。过十多处。 顿号、用于句子内部并列词语之间的停顿。

2时间介词in,on,at的用法

介词in on at 表示时间的用法及区别 Step1 Teaching Aims 教学生掌握时间介词in,on和at的区别及用法。 Step2 Teaching Key and Difficult Points 教学生掌握时间介词in,on和at的区别及用法。 Step3 Teaching Procedures 1.用in的场合后所接的都是较长时间 (1)表示“在某世纪/某年代/特定世纪某年代/年/季节/月”这个含义时,须用介词in Eg: This machine was invented in the eighteenth century. 这台机器是在18世纪发明的。 、 She came to this city in 1980. 他于1980年来到这个城市。 It often rains here in summer. 夏天这里常常下雨。 (2)表示“从现在起一段时间以后”时,须用介词in。(in+段时间表将来) Eg: They will go to see you in a week. 他们将在一周后去看望你。

I will be back in a month. 我将在一个月后回来。 (3)泛指一般意义的上、下午、晚上用in, in the morning / evening / afternoon Eg: They sometimes play games in the afternoon. 他们有时在下午做游戏。 Don't watch TV too much in the evening. 晚上看电视不要太多。(4)A. 当morning, evening, afternoon被of短语修饰,习惯上应用on, 而不用in. Eg: on the afternoon of August 1st & B. 但若前面的修饰词是early, late时,虽有of短语修饰,习惯上应用in, 而不用on. Eg: in the early morning of September 10th 在9月10的清晨; Early in the morning of National Day, I got up to catch the first bus to the zoo. 国庆节一清早,我便起床去赶到动物园的第一班公共汽车。 2.用on的场合后所接的时间多与日期有关 (1)表示“在具体的某一天”或(在具体的某一天的)早上、中午、晚上”,或“在某一天或某一天的上午,下午,晚上”等,须用介

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at 表示某个具体时刻。 at eight o’clock 在8点钟 at this time of the year 在一年中的这个时候 at the moment 在那一时刻 at that time 在那时 注意:在英语中,如果时间名词前用this, last, next 等修饰时,像这样的表示,“在某时”的时间短语前,并不需要任何介词。 例如:last month, last week, this year, this week, next year, the next day, the next year 等。 1.What’s the weather like in spring/summer/autumn/winter in your country? 你们国家春天/夏天/秋天/冬天的天气怎么样? in 在年、月、周较长时间内 in a week 在里面 in the room 用某种语言 in English 穿着 in red on 某日、某日的上下午on Sunday afternoon 在……上面 on the desk 靠吃……为生live on rice 关于 a book on Physics 〔误〕We got to the top of the mountain in daybreak. 〔正〕We got to the top of the mountain at day break. 〔析〕at用于具体时刻之前,如:sunrise, midday, noon, sunset, midnight, night。〔误〕Don't sleep at daytime 〔正〕Don't sleep in daytime. 〔析〕in 要用于较长的一段时间之内,如:in the morning / afternoon, 或in the week / month / year. 或in spring / supper /autumn / winter等等。 〔误〕We visited the old man in Sunday afternoon. 〔正〕We visited the old man on Sunday afternoon. 〔析〕in the morning, in the afternoon 如果在这两个短语中加入任何修饰词其前面的介

inonat的时间用法和地点用法版

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日期、onJune2nd在六月二日 onthesecond(ofJune2nd)在六月的第二天即在六月二日onthemorningofJune2nd在六月二日的早晨,onarainymorning在一个多雨的早晨 onacertainday在某天 onthesecondday在第二天(以过去某天为参照) 关于 In 1 2) InJune在六月 inJune,2010在2010年六月 in2010在2010年 inamonth/year在一个月/年里(在将来时里翻译成一个月/年之后) inspring在春天

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inonat的时间用法和地点用法版

i n o n a t的时间用法和 地点用法版 集团档案编码:[YTTR-YTPT28-YTNTL98-UYTYNN08]

i n,o n,a t的时间用法和地点用法 一、in,on,at的时间用法 1、固定短语: inthemorning/afternoon/evening在早晨/下午/傍晚, atnoon/night在中午/夜晚,(不强调范围,强调的话用duringthenight)earlyinthemorning=intheearlymorning在大清早, lateatnight在深夜 ontheweekend在周末(英式用attheweekend在周末,atweekends每逢周末)onweekdays/weekends在工作日/周末, onschooldays/nights在上学日/上学的当天晚上, 2、不加介词 this,that,last,next,every,one,yesterday,today,tomorrow,tonight,all,most等之前一般不加介词。如, thismorning今天早晨 (on)thatday在那天(thatday更常用些) lastweek上周 nextyear明年 thenextmonth第二个月(以过去为起点的第二个月,nextmonth以现在为起点的下个月) everyday每天 onemorning一天早晨 yesterdayafternoon昨天下午 tomorrowmorning明天早晨

allday/morning/night整天/整个早晨/整晚(等于 thewholeday/morning/night) mostofthetime(在)大多数时间 3、一般规则 除了前两点特殊用法之外,其他≤一天,用on,>一天用in,在具体时刻或在某时用at(不强调时间范围) 关于on On指时间表示: 1)具体的时日和一个特定的时间,如某日,某节日,星期几等。Hewillcometomeetusonourarrival. OnMay4th(OnSunday,OnNewYear’sday,OnChristmasDay),therewillbeacelebra tion. 2)在某个特定的早晨,下午或晚上。 Hearrivedat10o’clocko nthenightofthe5th. Hediedontheeveofvictory. 3)准时,按时。 Iftherainshouldbeontime,Ishouldreachhomebeforedark. 生日、onmyninthbirthday在我九岁生日那天 节日、onTeachers’Day在教师节 (注意:节日里有表人的词汇先复数再加s’所有格,如 onChildren’sDay,onWomen’sDay,onTeachers’Day有四个节日强调单数之意思, onMother’sDay,onFather’sDay,onAprilFool’sDay,onValentine’sDay)星期、onSunday在周日,onSundaymorning在周日早晨

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Don't watch TV too much in the evening. 晚上看电视不要太多。 (4)A. 当morning, evening, afternoon被of短语修饰,习惯上应用on, 而不用in. Eg: on the afternoon of August 1st (5)B. 但若前面的修饰词是early, late时,虽有of短语修饰,习惯上应用in, 而不用on. Eg: in the early morning of September 10th 在9月10的清晨; in the late afternoon of September 12th 在9月12日的傍晚。 Early in the morning of National Day, I got up to catch the first bus to the zoo. 国庆节一清早,我便起床去赶到动物园的第一班公共汽车。 用on的场合后所接的时间多与日期有关 (1)表示“在具体的某一天”或(在具体的某一天的)早上、中午、晚上”,或“在某一天或 某一天的上午,下午,晚上”等,须用介词on。 Eg: Jack was born on May 10th, 1982. 杰克生于1982年5月10日。 They left on a rainy morning. 他们是在一个雨天的早上离开的。 He went back to America on a summer afternoon. 他于一个夏天的下午返回了美国。

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