Bogomol'nyi–Prasad–Sommerfield bound: Difference between revisions
Jump to navigation
Jump to search
en>Lockley m replace context tag with technical tag |
en>ChrisGualtieri m →References: Remove stub tag(s). Page is start class or higher + General Fixes + Checkwiki fixes using AWB |
||
| Line 1: | Line 1: | ||
In [[combinatorics]], the '''Erdős–Ko–Rado theorem''' of [[Paul Erdős]], [[Chao Ko]], and [[Richard Rado]] is a theorem on [[Extremal set theory#Intersecting families|intersecting set families]]. It is part of the theory of [[hypergraph]]s, specifically, uniform hypergraphs of rank ''r''. | |||
The theorem is as follows. If <math>n\geq2r</math> and <math>A</math> is a family of distinct [[subset]]s of <math>\{1,2,...,n\}</math> such that each subset is of size <math>r</math> and each pair of subsets [[Intersection (set theory)|intersects]], then the maximum number of sets that can be in <math>A</math> is given by the [[binomial coefficient]] | |||
:<math>\binom{n-1}{r-1}.</math> | |||
(Since a family of sets may be called a hypergraph, and since every set in <math>A</math> has size ''r'', <math>A</math> is a uniform hypergraph of rank ''r''.) | |||
According to {{harvtxt|Erdős|1987}} the theorem was proved in 1938, but was not published until 1961 in an apparently more general form. The subsets in question were only required to be size ''at most'' <math>r</math>, and with the additional requirement that no subset be contained in any other. This statement is not actually more general: any subset that has size less than <math>r</math> can be increased to size <math>r</math> to make the above statement apply. | |||
==Proof== | |||
The original proof of 1961 used [[mathematical induction|induction]] on ''n''. In 1972, [[Gyula O. H. Katona]] gave the following short proof using [[double counting (proof technique)|double counting]]. | |||
Suppose we have some such family of subsets ''A''. Arrange the elements of {1, 2, ..., ''n''} in any [[cyclic order]], and consider the sets from ''A'' that form intervals of length ''r'' within this cyclic order. For example if ''n'' = 8 and ''r'' = 3, we could arrange the numbers {1, 2, ..., 8} into the cyclic order (3,1,5,4,2,7,6,8), which has eight intervals: | |||
:(3,1,5), (1,5,4), (5,4,2), (4,2,7), (2,7,6), (7,6,8), (6,8,3), and (8,3,1). | |||
However, it is not possible for all of the intervals of the cyclic order to belong to ''A'', because some pairs of them are disjoint. Katona's key observation is that at most ''r'' of the intervals for a single cyclic order may belong to ''A''. To see this, note that if (''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''r''</sub>) is one of these intervals in ''A'', then every other interval of the same cyclic order that belongs to ''A'' separates ''a<sub>i</sub>'' and ''a''<sub>''i'' + 1</sub> for some ''i'' (that is, it contains precisely one of these two elements). The two intervals that separate these elements are disjoint, so at most one of them can belong to ''A''. Thus, the number of intervals in ''A'' is one plus the number of separated pairs, which is at most ''(r-1)''. | |||
Based on this idea, we may count the number of pairs (''S'',''C''), where ''S'' is a set in ''A'' and ''C'' is a cyclic order for which ''S'' is an interval, in two ways. First, for each set ''S'' one may generate ''C'' by choosing one of ''r''! permutations of ''S'' and (''n'' − ''r'')! permutations of the remaining elements, showing that the number of pairs is |''A''|''r''!(''n'' − ''r'')!. And second, there are (''n'' − 1)! cyclic orders, each of which has at most ''r'' intervals of ''A'', so the number of pairs is at most r(''n'' − 1)!. Combining these two counts gives the inequality | |||
:<math>|A|r!(n-r)!\le r(n-1)!</math> | |||
and dividing both sides by ''r''!(''n'' − ''r'')! gives the result | |||
:<math>|A|\le \frac{r(n-1)!}{r!(n-r)!}={n-1\choose r-1}.</math> | |||
==Families of maximum size== | |||
There are two different and straightforward constructions for an intersecting family of ''r''-element sets achieving the Erdős–Ko–Rado bound on cardinality. | |||
First, choose any fixed element ''x'', and let ''A'' consist of all ''r''-subsets of <math>\{1,2,...,n\}</math> that include ''x''. For instance, if ''n'' = 4, ''r'' = 2, and ''x'' = 1, this produces the family of three 2-sets | |||
:{1,2}, {1,3}, {1,4}. | |||
Any two sets in this family intersect, because they both include ''x''. | |||
Second, when ''n'' = 2''r'' and with ''x'' as above, let ''A'' consist of all ''r''-subsets of <math>\{1,2,...,n\}</math> that do not include ''x''. | |||
For the same parameters as above, this produces the family | |||
:{2,3}, {2,4}, {3,4}. | |||
Any two sets in this family have a total of 2''r'' = ''n'' elements among them, chosen from the ''n'' − 1 elements that are unequal to ''x'', so by the [[pigeonhole principle]] they must have an element in common. | |||
When ''n'' > 2''r'', families of the first type (variously known as sunflowers, stars, dictatorships, centred families, principal families) are the unique maximum families. {{harvtxt|Friedgut|2008}} proved that in this case, a family which is almost of maximum size has an element which is common to almost all of its sets. This property is known as ''stability''. | |||
==Maximal intersecting families== | |||
An intersecting family of ''r''-element sets may be maximal, in that no further set can be added without destroying the intersection property, but not of maximum size. An example with ''n'' = 7 and ''r'' = 3 is the set of 7 lines of the [[Fano plane]], much less than the Erdős–Ko–Rado bound of 15. | |||
==References== | |||
* {{citation | |||
| first = P. | last = Erdős | authorlink = Paul Erdős | |||
| contribution = My joint work with Richard Rado | |||
| editor-first = C. | editor-last = Whitehead | |||
| title = Surveys in combinatorics, 1987: Invited Papers for the Eleventh British Combinatorial Conference | |||
| series = London Mathematical Society Lecture Note Series | |||
| volume = 123 | |||
| publisher = Cambridge University Press | |||
| year = 1987 | |||
| pages = 53–80 | |||
| isbn = 978-0-521-34805-8 | |||
| url = http://www.renyi.hu/~p_erdos/1987-12.pdf | |||
}}. | |||
* {{Citation | |||
| first1 = P. | last1 = Erdős | author1-link = Paul Erdős | |||
| first2 = C. | last2 = Ko | author2-link = Chao Ko | |||
| first3 = R. | last3 = Rado | author3-link = Richard Rado | |||
| title = Intersection theorems for systems of finite sets | |||
| journal = The Quarterly Journal of Mathematics. Oxford. Second Series | |||
| volume = 12 | |||
| year = 1961 | |||
| pages = 313–320 | |||
| url = http://www.renyi.hu/~p_erdos/1961-07.pdf | |||
| doi = 10.1093/qmath/12.1.313 | |||
}}. | |||
*{{citation |first=Ehud|last=Friedgut |title=On the measure of intersecting families, uniqueness and stability |journal=Combinatorica |volume=28 |issue=5 |pages=503–528 |year=2008| url=http://www.ma.huji.ac.il/~ehudf/docs/X%5E2=0.pdf |doi=10.1007/s00493-008-2318-9}} | |||
* {{citation | |||
| first = G. O. H. | last = Katona | authorlink = Gyula O. H. Katona | |||
| title = A simple proof of the Erdös-Chao Ko-Rado theorem<!--Don't change the title of the Katona paper! Erdös is how it appeared.--> | |||
| journal = [[Journal of Combinatorial Theory, Series B]] | |||
| volume = 13 | |||
| year = 1972 | |||
| pages = 183–184 | |||
| doi = 10.1016/0095-8956(72)90054-8 | |||
| issue = 2 | |||
}}. | |||
{{DEFAULTSORT:Erdos-Ko-Rado theorem}} | |||
[[Category:Set families]] | |||
[[Category:Theorems in discrete mathematics]] | |||
[[Category:Articles containing proofs]] | |||
[[Category:Factorial and binomial topics]] | |||
[[Category:Paul Erdős|Ko-Rado theorem]] | |||
Revision as of 19:32, 12 December 2013
In combinatorics, the Erdős–Ko–Rado theorem of Paul Erdős, Chao Ko, and Richard Rado is a theorem on intersecting set families. It is part of the theory of hypergraphs, specifically, uniform hypergraphs of rank r.
The theorem is as follows. If and is a family of distinct subsets of such that each subset is of size and each pair of subsets intersects, then the maximum number of sets that can be in is given by the binomial coefficient
(Since a family of sets may be called a hypergraph, and since every set in has size r, is a uniform hypergraph of rank r.)
According to Template:Harvtxt the theorem was proved in 1938, but was not published until 1961 in an apparently more general form. The subsets in question were only required to be size at most , and with the additional requirement that no subset be contained in any other. This statement is not actually more general: any subset that has size less than can be increased to size to make the above statement apply.
Proof
The original proof of 1961 used induction on n. In 1972, Gyula O. H. Katona gave the following short proof using double counting.
Suppose we have some such family of subsets A. Arrange the elements of {1, 2, ..., n} in any cyclic order, and consider the sets from A that form intervals of length r within this cyclic order. For example if n = 8 and r = 3, we could arrange the numbers {1, 2, ..., 8} into the cyclic order (3,1,5,4,2,7,6,8), which has eight intervals:
- (3,1,5), (1,5,4), (5,4,2), (4,2,7), (2,7,6), (7,6,8), (6,8,3), and (8,3,1).
However, it is not possible for all of the intervals of the cyclic order to belong to A, because some pairs of them are disjoint. Katona's key observation is that at most r of the intervals for a single cyclic order may belong to A. To see this, note that if (a1, a2, ..., ar) is one of these intervals in A, then every other interval of the same cyclic order that belongs to A separates ai and ai + 1 for some i (that is, it contains precisely one of these two elements). The two intervals that separate these elements are disjoint, so at most one of them can belong to A. Thus, the number of intervals in A is one plus the number of separated pairs, which is at most (r-1).
Based on this idea, we may count the number of pairs (S,C), where S is a set in A and C is a cyclic order for which S is an interval, in two ways. First, for each set S one may generate C by choosing one of r! permutations of S and (n − r)! permutations of the remaining elements, showing that the number of pairs is |A|r!(n − r)!. And second, there are (n − 1)! cyclic orders, each of which has at most r intervals of A, so the number of pairs is at most r(n − 1)!. Combining these two counts gives the inequality
and dividing both sides by r!(n − r)! gives the result
Families of maximum size
There are two different and straightforward constructions for an intersecting family of r-element sets achieving the Erdős–Ko–Rado bound on cardinality. First, choose any fixed element x, and let A consist of all r-subsets of that include x. For instance, if n = 4, r = 2, and x = 1, this produces the family of three 2-sets
- {1,2}, {1,3}, {1,4}.
Any two sets in this family intersect, because they both include x. Second, when n = 2r and with x as above, let A consist of all r-subsets of that do not include x. For the same parameters as above, this produces the family
- {2,3}, {2,4}, {3,4}.
Any two sets in this family have a total of 2r = n elements among them, chosen from the n − 1 elements that are unequal to x, so by the pigeonhole principle they must have an element in common.
When n > 2r, families of the first type (variously known as sunflowers, stars, dictatorships, centred families, principal families) are the unique maximum families. Template:Harvtxt proved that in this case, a family which is almost of maximum size has an element which is common to almost all of its sets. This property is known as stability.
Maximal intersecting families
An intersecting family of r-element sets may be maximal, in that no further set can be added without destroying the intersection property, but not of maximum size. An example with n = 7 and r = 3 is the set of 7 lines of the Fano plane, much less than the Erdős–Ko–Rado bound of 15.
References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.