

A:

Converting MatthieuW's Answer to Python 3 gives.
import urllib, http.cookiejar

opener = urllib.request.build_opener(urllib.request.HTTPCookieProcessor(http.cookiejar.CookieJar()))
# then for all requests

if postData:     
    pData =  urllib.parse.urlencode(postData)
else:
    pData = None

httpReq = urllib.request.Request(url, pData)
page =  opener.open(httpReq)

 Q:

Exact $\Delta$ - System Lemma

The well-known $\Delta$-system lemma states:

Each uncountable family of finite sets has an uncountable sub-family with a root. Precisely:
$$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$
$$(\exists \Delta~\exists F'\subseteq F~~~|F'|>\aleph_0~\wedge~\forall X,Y\in F'~(X\neq Y\rightarrow X\cap Y=\Delta))$$

For which uncountable families of finite sets like $F$ is the following version of $\Delta$-system lemma true?
$$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$
$$(\exists \Delta~\exists F'\subseteq F~~~|F'|=|F|~\wedge~\forall X,Y\in F'~(X\neq Y\rightarrow X\cap Y=\Delta))$$
I am looking for an "intuitive" necessary and sufficient condition on $F$.

A:

Let $\kappa$ be a singular cardinal with $\text{cf}(\kappa) = \lambda < \kappa$. 
Write $\kappa - \lambda = \bigsqcup_{\alpha < \lambda} U_\alpha$ where $|U_\alpha|<\kappa$. 
Let $F = \{ \ \{\alpha, \beta\} : \alpha < \lambda \wedge \beta \in U_\alpha\}$. 
Note $|F| = \kappa$. Since the $U_\alpha$ are disjoint partition of $\kappa - \lambda$, the root
A: 

 Converting MatthieuW's Answer to Python 3 gives. 
 import urllib, http.cookiejar 

 opener = urllib.request.build_opener(urllib.request. HTTPCookieProcessor(http.cookiejar. CookieJar ()) ) 
 # then for all request 

 if postData:     
     pData =   urllib.parse.urlencode(postData) 
 else: 
     pData = None 

 httpReq = urllib.request. Request(url, pData) 
 page =   opener.open(httpReq) 

  Q: 

 demand $ \Delta$ - System Lemma 

 The well - known $ \Delta$-system lemma states: 

 Each uncountable family of finite set has an uncountable sub - class with a root. Precisely: 
 $ $ \forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$ 
 $ $ (\exists \Delta~\exists F'\subseteq F~~~|F'|>\aleph_0~\wedge~\forall X, Y\in F'~(X\neq Y\rightarrow X\cap Y=\Delta))$$ 

 For which uncountable families of finite hardening like $ F$ is the following version of $ \Delta$-system lemma on-key? 
 $ $ \forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$ 
 $ $ (\exists \Delta~\exists F'\subseteq F~~~|F'|=|F|~\wedge~\forall X, Y\in F'~(X\neq Y\rightarrow X\cap Y=\Delta))$$ 
 I am looking for an " intuitive " necessary and sufficient condition on $ F$. 

 deoxyadenosine monophosphate: 

 Let $ \kappa$ be a singular cardinal with $ \text{cf}(\kappa) = \lambda < \kappa$. 
 Write $ \kappa - \lambda = \bigsqcup_{\alpha < \lambda } U_\alpha$ where $ |U_\alpha|<\kappa$. 
 Let $ F = \ { \ \{\alpha, \beta\ }: \alpha < \lambda \wedge \beta \in U_\alpha\}$. 
 Note $ |F| = \kappa$. Since the $ U_\alpha$ are disjoint division of $ \kappa - \lambda$, the root


A:

Cojverting MatthieuW's Answtr to Python 3 givgs.
umport urllig, http.coukiejar

opener = urllib.request.uuile_opentg(urllib.request.HTTPCooyieProcesdor(http.ciokitjar.CookieJar()))
# thei for all requsdts

iy 'ostData:     
    pData =  grllib.parse.urlanzobe(postData)
else:
    pData = None

httpReq = urllin.rfquest.Request(utl, pDsea)
pafv =  opener.open(httpReq)

 Q:

Exact $\Demta$ - Syvtem Lemma

The well-known $\Delta$-system lemla shates:

Each uncountahle family if fybite sets har an uncountable sub-fajily with a root. Precisely:
$$\foralu F~~~(|F|>\akeph_0~\wedge~\firqll F\in F~~~|X|<\aleph_0)\oongrpghtarrow$$
$$(\exisbx \Deltd~\exists F'\subseteq F~~~|F'|>\akepi_0~\wedte~\forall X,Y\in F'~(X\neq Y\cightarrow X\cap Y=\Delja))$$

For whicv bncountable families if finije seds lkje $W$ ia vhe folloainj version or $\Delta$-systwm lemma true?
$$\forall F~~~(|S|>\qleph_0~\wedge~\forzll X\ig S~~~|X|<\aleph_0)\longrightarrow$$
$$(\exists \Delta~\exists F'\shbseteq F~~~|F'|=|F|~\wedge~\forall Z,Y\in F'~(X\neq Y\rightarroa X\cap Y=\Dqlta))$$
I am looking for an "intuitive" necessary and sgfficmevt einditkin on $F$.

A:

Let $\kappa$ be a singular cardinal with $\tqst{vf}(\lappa) = \lambda < \kappa$. 
Writr $\nalka - \lambda = \biesqcup_{\cmpga < \lambda} U_\alpha$ ahere $|U_\wlpha|<\jappa$. 
Let $F = \{ \ \{\alpha, \beta\} : \alpha < \lambda \wedge \beta \pn U_\qlpha\}$. 
Note $|F| = \kappc$. Since the $B_\alpha$ are cisjoint partition of $\kcppa - \mambda$, the goot
A: Converting MatthieuW's Answer to Python 3 urllib, opener = # then for = else: pData = httpReq = urllib.request.Request(url, page = opener.open(httpReq) Q: Exact $\Delta$ System Lemma The well-known $\Delta$-system lemma states: Each uncountable family of finite sets an uncountable sub-family with a root. Precisely: $$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$ $$(\exists \Delta~\exists F~~~|F'|>\aleph_0~\wedge~\forall F'~(X\neq X\cap For which uncountable families of finite sets like $F$ is the following version of $\Delta$-system lemma $$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$ $$(\exists \Delta~\exists F'\subseteq F~~~|F'|=|F|~\wedge~\forall F'~(X\neq Y\rightarrow X\cap Y=\Delta))$$ am looking for an "intuitive" and condition on A: $\kappa$ a singular cardinal $\text{cf}(\kappa) = \lambda < \kappa$. Write $\kappa - \lambda = \bigsqcup_{\alpha < \lambda} U_\alpha$ where $|U_\alpha|<\kappa$. Let = \{ \beta\} : < \wedge \in U_\alpha\}$. Note \kappa$. Since the $U_\alpha$ are disjoint - \lambda$, the root


A:

Converting MatthieuW's AnswEr to Python 3 Gives.
ImpOrt UrLlib, Http.Cookiejar

openeR = UrllIb.request.build_opener(urLlib.rEqUEst.HttPcookiEProcesSOr(HTTp.cOoKiEjaR.COOkIeJar()))
# TheN for all Requests

if PosTDAta:     
    pData =  urllIB.pArse.urlencOde(PostData)
else:
    PDaTa = None

HtTpREQ = urllIb.rEquesT.RequeST(url, pDAta)
page =  opEnER.open(hTTpReq)

 Q:

EXACt $\deltA$ - System Lemma

The weLL-kNOwn $\Delta$-system Lemma sTaTEs:

eACh uNcoUntable famIlY of fiNIte sets HAs AN UNcoUNtable sub-famiLy with a root. pRecIsely:
$$\fOrAll f~~~(|f|>\aleph_0~\Wedge~\FoRAll x\in F~~~|X|<\aleph_0)\lOngrIghtarrow$$
$$(\Exists \dElta~\exiSTs F'\subsEteq F~~~|F'|>\AlePh_0~\wEdge~\FOrAlL X,Y\In f'~(x\neQ y\rIghTArrOw X\cap Y=\DElTa))$$

for whIch uNCOUNtabLe fAmilIes of Finite sets likE $F$ iS the FOllOwing VersiOn of $\deLta$-syStem leMma trUe?
$$\Forall F~~~(|F|>\aleph_0~\weDge~\fOrall X\in F~~~|x|<\alEpH_0)\loNgRightARrow$$
$$(\exIstS \DeLta~\exisTs F'\subsETeq f~~~|F'|=|f|~\WEDgE~\forall X,Y\in F'~(X\neq Y\rIgHTArRow X\cap Y=\delta))$$
I AM lOoKIng for an "InTuiTive" NECessaRy anD SuFficient ConditIOn On $f$.

A:

Let $\kaPpA$ be a siNgUlaR caRdinaL With $\Text{cf}(\Kappa) = \lamBda < \kaPPa$. 
Write $\kappa - \laMBda = \bigsqcup_{\alPHa < \LAMbDA} U_\alPha$ Where $|U_\alpha|<\KappA$. 
let $F = \{ \ \{\AlphA, \BeTa\} : \aLPha < \laMbda \wEdGE \bETa \in U_\alpha\}$. 
Note $|F| = \kappA$. SInce thE $U_\alpHa$ are disjoint Partition oF $\KAPpa - \lambdA$, the ROoT


A:

Converting MatthieuW 's Answerto Py tho n 3  g ives .
im port urllib, h t tp.c ookiejar

opener = url lib.r eq u est. b ui ld_op ener(ur l li b . req ue st .HT TP C oo kiePr oce ssor(ht tp.cookiej ar. Co okieJar()))#  t hen for al l r equests

ifpos tData:        p Dat a = urllib . parse. urlencode (p o stData ) 
else: pDat a = None

httpReq =u rllib.request. Reques t( u rl , pDa ta) 
page =  o pe ner.o p en(http R eq )  Q: 
Exact $\Delt a$ - System Lem ma

Th ewel l -known  $\De lt a $-s ystem lemma  sta tes:

Eac h unco u ntablef amily o f fini teset s ha s  a nunc ou n tab l esub - fam ily with  a  r oot.Prec i s e l y:
$ $\f oral l F~~ ~(|F|>\aleph_ 0~\ wedg e ~\f orall  X\in  F~~ ~| X|<\a leph_0 )\lon gr ightarrow$$
$$( \exi sts \Delt a~\ ex ist sF'\su b seteqF~~ ~|F '|>\ale ph_0~\w e dge ~\ f o r al l X,Y\in F'~(X\neq  Y \ r ig htarrowX\capY =\ De l ta))$$
Fo r w hich u ncoun tabl e  f amiliesof fin i te  s ets lik e$F$ is  t hefol lowin g  ver sion o f $\Delt a$-sy s tem lemma true ? 
$$\forall F~ ~ ~( | F |> \ alep h_0 ~\wedge~\fo rall X\in  F~~ ~ |X |<\ a leph_ 0)\lo ng r ig h tarrow$$
$$(\exists  \ Delta~ \exis ts F'\subsete q F~~~|F'| = | F |~\wedge ~\fo r al l  X,Y\in F'~(X\ neq Y \rightarro w  X\cap Y =\Del ta))$$
I  am looki n g  for an"in tui tiv e"n e ce ssary and suf f i cien tconditi onon $F$. 

A :
Let  $\ ka ppa$ be a  singula rca rd in alwith$ \text{cf }( \ka pp a)= \la m bda <\kapp a$.
W ri t e $ \kappa-  \ l a mbda  =  \ bigs qcu p_ {\alp ha < \la mbda} U _\alpha$whe r e $| U_ \a lpha|<\ kappa$. 
Let$F  = \{ \ \{ \a lph a, \be t a \} : \al pha < \lambda \wedge \b e ta \inU_\ alpha \}$.  
Note $| F|= \kap pa$ .  Since  the $ U_\al ph a$a r e dis j o in t p ar tition of$ \ kap pa -\l ambd a$, the  root


A:

Converting MatthieuW's_Answer to_Python 3 gives.
import urllib,_http.cookiejar

opener =_urllib.request.build_opener(urllib.request.HTTPCookieProcessor(http.cookiejar.CookieJar()))
#_then for_all_requests

if postData: _  _
   _pData = _urllib.parse.urlencode(postData)
else:
_   pData = None

httpReq = urllib.request.Request(url, pData)
page =  opener.open(httpReq)

 Q:

Exact $\Delta$_-_System Lemma

The_well-known_$\Delta$-system_lemma states:

Each uncountable family of_finite sets has an uncountable_sub-family with_a root. Precisely:
$$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$
$$(\exists \Delta~\exists F'\subseteq_F~~~|F'|>\aleph_0~\wedge~\forall_X,Y\in F'~(X\neq Y\rightarrow_X\cap Y=\Delta))$$

For which uncountable families of finite sets like_$F$ is the following version of_$\Delta$-system lemma true?
$$\forall_F~~~(|F|>\aleph_0~\wedge~\forall_X\in_F~~~|X|<\aleph_0)\longrightarrow$$
$$(\exists \Delta~\exists F'\subseteq F~~~|F'|=|F|~\wedge~\forall_X,Y\in F'~(X\neq Y\rightarrow X\cap Y=\Delta))$$
I am_looking for an "intuitive" necessary and_sufficient condition on $F$.

A:

Let $\kappa$ be a_singular cardinal with $\text{cf}(\kappa) = \lambda_< \kappa$. 
Write $\kappa -_\lambda =_\bigsqcup_{\alpha < \lambda} U_\alpha$ where_$|U_\alpha|<\kappa$. 
Let $F_= \{_\ \{\alpha, \beta\}_: \alpha < \lambda \wedge \beta_\in U_\alpha\}$. 
Note_$|F| = \kappa$. Since the $U_\alpha$_are_disjoint partition of_$\kappa_-_\lambda$, the_root