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I have discovered a truly remarkable description of myself that this margin is too small to contain.
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comment 
independent subset problems
(In particular, your problem seems to include 3dimensional matching as a special case, so it is NPcomplete.) 
Nov 19 
comment 
independent subset problems
I don't see what purpose the function $f$ serves here. Your problem seems to be equivalent to looking for a matching in the hypergraph with edge set $f(N)$, so why not work with the hypergraph directly? 
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answered  Deciding whether a given graph has an ffactor or not! 
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comment 
What is the correct statement of this ErdösGallaiTuza problem generalizing Turan's triangle theorem?
@darijgrinberg: Thanks for your comments! Point taken regarding (1). As far as Lemma 2.1 goes, nothing is really gained by allowing S to be empty or to be all of V(G) (this only yields f_1(G) <= f_1(G) ). It seems a hair simpler to just keep the same hypothesis as Corollary 2.2, where the condition really is needed. 
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answered  What is the correct statement of this ErdösGallaiTuza problem generalizing Turan's triangle theorem? 
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