If you start from the lines of the affine plane for GF(5), you get 30 teams of 5 players each, chosen from 25 players: 

ABCDE AFKPU AGMSY AHOQX AILTW AJNRV BFOSW BGLQV BHNTU BIKRY BJMPX CFNQ
+Y CGKTX CHMRW CIOPV CJLSU DFMTV DGORU DHLPY DINSX DJKQW EFLRX EGNPW E
+HKSV EIMQU EJOTY FGHIJ KLMNO PQRST UVWXY

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If you omit the last player from each team (thus always omitting Y from the teams ey is in), you get the following 30 teams. 

ABCD AFKP AGMS AHOQ AILT AJNR BFOS BGLQ BHNT BIKR BJMP CFNQ CGKT CHMR 
+CIOP CJLS DFMT DGOR DHLP DINS DJKQ EFLR EGNP EHKS EIMQ EJOT FGHI KLMN
+ PQRS UVWX

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This is minimal maximal in that you cannot add a new team to it, but if you remove the team UVWX, you can add several more. If you keep adding the (lexicographically) first team that goes with the rest, you get these 35 teams: 

ABCD AFKP AGMS AHOQ AILT AJNR BFOS BGLQ BHNT BIKR BJMP CFNQ CGKT CHMR 
+CIOP CJLS DFMT DGOR DHLP DINS DJKQ EFLR EGNP EHKS EIMQ EJOT FGHI KLMN
+ PQRS 
AEUV BEWX FJUW GJVX KOUX LOVW

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Update: Similarly, you also get 35 teams if you start from nothing and keep adding the (lexicographically) first team that doesn't intersect any chosen team in two or more members (Update: this is the same result as ketema's got): 

ABCD AEFG AHIJ AKLM ANOP AQRS ATUV BEHK BFIL BGJM BNQT BORU BPSV CEIM 
+CFHN CGKO CJLP CQUW CRTX DEJN DFKP DGHL DIOQ DMRV DSTW ELOS EPQX FJOT
+ FMSU FVWX GINR HMOW IKSX JKQV LNUX

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The J source code to generate this is (takes several seconds to run): 

   players =: a.{~65+i.24
   teams =: (#~]-:"1~."1)~./:~/:~"1>,{4$<players
   good =: teams#~[:(*./"1) 1>: teams&(+/@:e."1"1 _)
   ,,&' '"1 (,{.@:good)^:35 i.0 4

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In reply to Re: Unique Combos with Math::Combinatorics by ambrus
in thread Unique Combos with Math::Combinatorics by ketema

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