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ErdsSelfridge
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The potential method in Positional games (From 'TicTacToe Theory' by Jozsef Beck) ... ??? ??? ?? ?????? ???? ?? ???? ???? ?? ???????? - n - ???? ??? n-????????? ... – PowerPoint PPT presentation
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Title: ErdsSelfridge
1
???? Erdös-Selfridge The potential method in
Positional games (From TicTacToe Theory by
Jozsef Beck)
2
???? ???? - ?????????
- V ????????
- E ?????????, ???? ?? ??-????? ?? ????????
- (??? ?? ??? ???????? ??? ?????? ??????)
- ??? ??? ?? ?????? ???? ?? ???? ???? ?? ???????? -
n - ???? ??? n-????????? - ????? ??? n-????????? ???, ?? ?????
3
???? ???? - ?????????
4
???? ???? - ?????????
- ???? ?? ??????? - ??? ?????? ?????? ????????
?????? ????? ????? ???????? ??????????? (????? n
???????? ???? ????? ???? ??? ???? ?? ???? ???). - ?????, ????? ?????, ????? ?? ????? ??? ????
maker/breaker - ??? maker, breaker ?????? ?? ??????
- ?? maker ????, ??? ??? ???????
5
) breaker ???? 1 (
- ???? ??? n-????????? G (F,V)
- ????
- Maxdeg(G) ????? ????? ????? ?? ????? ????? ??
?????? ??? - ?? F Maxdeg(G) lt 2n ??? Breaker ???? ????
???? ???? ???.
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F maxdeg(G) lt 2n
- ?????? (??? ??? ??????) ????? ???????? ????????
?? ????? (??? ?????? ??????) - ????? Di ????? ??? ?????? ?? ?"????" ?????? ??
???? ???? i ????? ?????? Breaker ???? ?maker ???
i ???????? x1xi ?breaker ??? 1i- ????????
y1yi-1 - ????? ???? ?? ??? ?????? ??? ?? Breaker ????
????? ???? "????" ??? ????????? ?????? Di
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- Si ????? ?????? ????? ???? ???? i ?"??????"
- Us ???? ??? s ?????? ???, ???? ?????????
??????? - Di ???????? ????? ?? ???? ???? ?????? breaker
- ???? ??
- ??? ????? ???? 2-n
- ??? ???? ???? 1
- ??? ?? ?????? ??? ???? ???? ?????
- Breaker ???? ???? ?? ?????? ??? ??? ????.
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- ?????? ?? Breaker ??????? ??????yi , ????? ?? ??
?????? ???? ???? (??? ??? ????) - ?????, breaker ????? ?? ????? ?
- ????? ??????? ?? ????? ??? ??? ???????
- ??? ?????, maker ????? ???? ???? ????? ????
?????? ??????? xi1. ???? ??? ??? ???? ?? xi1 ??
???? ????? ?????? ???? ????? ?????? ?? ??????.
?????, maker ?????
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- ???? ??
- breaker ??? ?? ??????? ?????? ?? ???? ??? ????
????? ?????? maker ???? ?? ???? ?????? ??????
breaker - ??? ??????? ??? ???? ????? ?? breaker ???? ??????
?? maker ????? ???? ??? ???? - ???? i1
0
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- ????? ?? D1 Dfirst (?????, ???? ????? ?????? ??
maker, ???? ????? ?????? ?? breaker) - ????? ??? ???
- Us n-1 ?????? ????? ?? x1
- Us n ?????? ????? ???? ?? x1
- ?????? 1 ??? ????? ?? ??? ??? ?maker ????? ????
????. ??? ????? ?????? ???? ???? ????? ???? ???
???? ?????? ?? ???? ?? ???.
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- ??? ???? F maxdeg(G) lt 2n ?????
- ????? ???????? ??????? ?? ?? ??????.
- n ??? ???? ???
- F 2n-1
- Maxdeg(G) 2n-1 ?????? ?????, ???? ???? ??
?????? - F maxdeg(G) 2n-1 2n-1 2n ?????
- ?? ?????, maker ???? ??? ???? ????? ????,
?Breaker ?? ??? ?? ????? ????? ???? - ????? ?????? ???? ?2n ????? ??? ?? ????
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) maker ???? 2 (
- ???? ?2(G) ????? ????? ????????? ??????? ?? ???
???????? ?-G. ?????, ????? ????????? ?? ?????
????? ?? ??? ????????. - ???? ??, ???? ???? ????, ???? ??? ????????
??????? ?? ???? ???, ?2 ???? ????. - ?Maker ?? ???????? ????? ???? ??
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- ????? ???? Ti, ???? ????????? ??????? ??? ????
???? ????? ?-i ?? ??? ??????? (?????, ???? ?maker
????? ?? ?????) - Us ???, ???? ????????? ??????? ???? S
- ???? ????? ????
- ????, ?? ?? ????????? ?????? ????????? ?? ??? ???
1 - ?? ?? ????????? ?????, ????????? ??? 2n
- ?? ?????? ???? ?????? ????, ?????? ?? ?????????
??? ?? 2
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- Maker ???? ????? ?? ?????? xi1 ?? ?????????
??????? ??? ???? ????? ??? ??? breaker ???? ?????
?????. ????? ??? ????? ?? ????????? ?? ?? ??????
????? ?? xi1 ???? ??? ???? ???? ????, ????
?breaker ???? ????? ????.
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- ???? ?????? ??? ????? xi1 ??? yi1 ???? ??????
?? ??? ?2 - ???? ????????? ??????? ??? ??? ???? ???? ????? 2
(?? ?????? ?? xi1,yi1) ????? Us2
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- ???? ?????? ??? ???? ????? ????????? ?? ???? ????
????
17
- ?????? ??? V ????????, ???? ??????? ???? ????
V
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??? ??? ?????? ???? maker?
- ??? ??? "?????" ?? ?? ?? ??? ???????? ?? maker
?????, ???? ?? ?? ?????? ?? breaker (?? ???
"???") - ????? ?? ??? i, ???? ????? ?-i ?? maker ??????
?-i-1 ?? breaker - ????? Us ???? ????????? ??????? ?? ??? maker
????????? ????
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- ????? ???? ????? ??? (yi, xi1) ???? ?????
??????? - ???? ????? ?? ?????? ?????? ???? ?? Yi
- ???? ????? ??????? ?????? ???? ?? Xi1
- ???? ????? ?????? ???? ??? ?????? ??? ?? maker
(Us2) ???? ?? xi1 ?? ????? ????? ???? ?????? - ???? ????? ??????? ???? ?? Xi1 ??? Yi ??????
?????
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- ??? ????? ???? ??????
- ???? i, maker ??? ???? i ???????? ????
- ??? ??? ?????????? ????, ?? ??? ????? ?2 ?????
??????? ?? xi1 - ???? ?????? ????? ????? ???? ???? ?-?2 i
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- ???? ?? ?????, ????? ??? ??? ??? ????????? ??
?????, Usn. ?????? ??? ????? ??? 2n - ???? ?maker ???? ???? ????? ?m-?. ????? Dm2n
- ??? Dfirst 0, ?? ??? ????? ????? ???? ?????
?????? - ????? 2n(m2-m)2?2
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- ?????, ?? ???-maker ?? ???????? ?????, ?????? ??
?2 ???, ??? ????? ???? ???????????? ????? ????.
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(No Transcript)
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