Transversals of hypergraphs with geometric flavor

Transversals of hypergraphs with geometric flavor

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!Z„ AsR${ ‚-s8}j‚ $R > A‚$t s ht$‚ RkR‚8 BP {Bt%‚- R‚R $t iQ  7‚%‚„sj $8}B„st {B8A$tsB„$sj „‚RZjR sABZ {Bt%‚- R‚R {st A‚ {B8}s{jk }Y„sR‚I $t Y‚ ‚„8$tBjBk `ZR $t„BIZ{‚I qY‚ b‚jjT,tBbt Y‚B„‚8 BP <‚jjk {st A‚ „‚TRs‚I sR + w>D  d bY‚t‚%‚„ "Q w>D c Q L d qY‚ VjBtu€j‚$8st }„BBP stI sAR„s{ stsjBZ‚R !Z„ I$R{ZRR$Bt 8s$tjk „‚%Bj%‚R s„BZtI Y‚ {‚j‚A„s‚I w€, RDTY‚B„‚8 BP VjBt stI €j‚$8st >nH ws RBjZ$Bt B Y‚ $R s RkR‚8 BP {Bt%‚- R‚R $t iQ b$Y "Rd w> D c € PB„ RB8‚ €  R  Q L d w$t BY‚„ bB„IR s8Bt stk € R‚R BP >  RB8‚ R $t‚„R‚{D Y‚t + w> D $R ABZtI‚I Ak

s PZt{$Bt BP Q € stI R yP Bt‚ $R tB $t‚„‚R‚I $t JZst$s$%‚ ABZtIR $ RZ^{‚R B {BtR$I‚„ Btjk R ) Q L d bY${Y b‚ IB P„B8 tBb Bt qY‚ }„BBP YsR bB 8s$t Rs‚R G‚ BZj$t‚ Y‚8 stI Rs‚ Y‚$„ sAR„s{ %‚„R$BtR 3$„R " w> D $R ABZtI‚I $t ‚„8R BP "Q w> D ZR$t Y‚ P„s{$Btsj <‚jjk Y‚B„‚8 BP €s{YsjR,$ stI N$Z N‚ r A‚ s }BRR$Ajk $tht$‚ R‚ RkR‚8 j‚ > A‚ st $t‚‚„ stI , # ; >(, dH G‚ Rsk Ys r YsR Y‚ P„s{$Btsj <‚jjk }„B}‚„k 3, , #D $P PB„ ) stk Dzi{YB${‚ BP R‚R Bd, B1, 3 3 3 , Bz ; r RZ{Y Ys 7;{ B7 W) ! PB„ PB„ s j‚sR

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k ( Y‚„‚ ‚-$RR # k ( RZ{Y Ys Y‚ RkR‚8 BP {Bt%‚- R‚R $t Q Rs$Rh‚R 3 A‚ s wht$‚D RkR‚8 BP {Bt%‚- R‚R b$Y "Q w> D c € b‚ bst B ABZtI " w> D N‚ jU > >(, dH A‚ s b‚$Y PZt{$Bt PB„ bY${Y " w> D $R ss$t‚I G‚ 8sk sRRZ8‚ Ys Y‚ %sjZ‚R BP j s„‚ „s$Btsj stI b„$‚ jwB D ) zE*  G‚ PB„8 s t‚b RkR‚8 > I Ak }Z$t zB {B}$‚R BP B $tB > I PB„ ‚s{Y B ; >  N‚ ] ) /> I / y RZ^{‚R B RYBb Ys Y‚„‚ $R s }B$t ? {B88Bt B s j‚sR #] R‚R BP > I PB„ RB8‚ # ) #wQ, €D k ( R$t{‚ Y‚t  d d  B ;> U ?;B jwB D ) E N /\B ; > I U ? ; B i/  # ]E ) #" w> D N‚ I <‚jjk }„B}‚„k $ RZ^{‚R B RYBb > ))\Bd , B1 , 3 3 3 , B] i Ak Y‚D P„s{$Btsj ] i Ys 7;{ B7 W) ! PB„ s j‚sR QLd $tI‚- R‚R { BP R$9‚ Q Ld qB Y$R ‚tI b‚ BAR‚„%‚ Ys "Q w> ID c €I ) Qw€  dD L d wR$t{‚ s8Bt stk €I R‚R BP > I ‚$Y‚„ Y‚ Rs8‚ R‚ $R „‚}‚s‚I Q L d $8‚R B„ Y‚„‚ s„‚ € I$R$t{ R‚R stI $t Y‚ js‚„ {sR‚ b‚ ZR‚ "Q w> D c €D 7B ‚%‚„k €ITZ}j‚ BP R‚R BP > I {Bt„$AZ‚R st $t‚„R‚{$t wQ L dDTZ}j‚ stI Y‚ s„Z8‚t $R ht$RY‚I Ak IBZAj‚ {BZt$t qY$R }„BBP $t sAR„s{ R‚$t k$‚jIRU 3B„ ‚%‚„k Q stI € Y‚„‚ ‚-$RR k ( RZ{Y Ys PB„ stk ht$‚ Ps8$jk > BP R‚R Rs$RPk$t 3 D c € b‚ Ys%‚ + w> D  x ) x w€, Q, #D

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qY‚ R‚{BtI Rs‚ BP Y‚ VjBtu€j‚$8st }„BBP RYBbR Ys + $R ABZtI‚I Ak s PZt{$Bt BP + PB„ ‚%‚„k Ps8$jk > BP {Bt%‚- R‚R $t Q  qY$R $R s Ps$„jk I$„‚{ {BtR‚JZ‚t{‚ BP s „‚RZj BP VjBt ?s„? stk 3[ Z„‚I$ stI €j‚$8st sABZ Y‚ RBT{sjj‚I b‚s, Tt‚R PB„ {Bt%‚- R‚R bY${Y $t Z„t $R AsR‚I Bt s R‚j‚{$Bt j‚88s BP ?s„?stkU PB„ ‚%‚„k zT}B$t R‚ h 2 Q $t ‚t‚„sj D z i}BR$$Bt Y‚„‚ ‚-$RR s }B$t {Bts$t‚I $t Y‚ {Bt%‚- YZjjR BP s j‚sR  QLd RZAR‚R < g h b$Y / I‚tB‚R Y‚ Ps8$jk BP Y‚ {Bt%‚YZjjR BP D i />/ sjj wQ L dDT}B$t RZAR‚R BP h  $ RZ^{‚R B %‚„$Pk Ys s j‚sR QLd BP Y‚

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wQ L dDTZ}j‚R BP R‚R $t > $t‚„R‚{ k IBZAj‚ {BZt$t Y$R PBjjBbR P„B8 1 Q {st A‚ q%‚„A‚„:R Y‚B„‚8  bY${Y $8}j$‚R Ys stk wQ L dD }B$tR $t }s„$$Bt‚I $tB Q L d ‚JZsjTR$9‚ R‚R bYBR‚ {Bt%‚- YZjj sjj Ys%‚ s {B88Bt }B$t k s R$8$js„ 8‚YBI st sAR„s{ stsjBZ‚ $R }„B%‚I >1HU 3B„ ‚%‚„k $t‚‚„ Q  d Y‚„‚ ‚-$RR k ( RZ{Y Ys Y‚ PBjjBb$t YBjIR N‚ > A‚ s ht$‚  ) \) E U E g >i A‚ Y‚ Ps8$jk BP sjj $t‚„R‚{$BtR Ps8$jk BP R‚R stI j‚ >  Rs$Rh‚R 3  yP > + w> D  d N + w> D  bY‚„‚ d stI 1 I‚}‚tI Btjk Bt Q stI #  qY‚ ,‚k t‚b R‚} $t Y‚ }„BBP $R I‚„$%$t s b‚s, q%‚„A‚„Tk}‚ „‚RZj P„B8 Y‚ P„s{$Btsj <‚jjk }„B}‚„k BP >  

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V 2 Q wb‚s, d0 Tt‚D BP R$9‚ s 8BR W wQ, 0D RZ{Y Ys stk {Bt%‚- g b$Y /g  … /  d0 /… / $t‚„R‚{R V  qY‚ }„BAj‚8 BP ‚R$8s$t W wQ, 0D YsR A‚‚t RZI$‚I $t R‚%‚„sj }s}‚„R AZ $ $R R$jj Ps„ P„B8 RBj%‚IU Y‚ A‚R Z}}‚„ ABZtI PB„ Q h-‚I $R 5w0Q wjB 0DLzX0wQD D stI tB tBt„$%$sj jBb‚„ ABZtI $$R ,tBbt wPB„ Q stI 0 h-‚I B s RZ$sAj‚ {BtRst s jBb‚„ ABZtI BP  w QD bsR $%‚t $t >HD yt Y‚ sAR„s{ R‚$t 8‚t$Bt‚I sAB%‚ ws RkR‚8 b$Y s QTN‚„sk t‚„%‚ B„ `ZR sRRZ8$t Y‚ P„s{$Btsj <‚jjk }„B}‚„k PB„ >  D Y‚ Z}}‚„ ABZtI Bt 3 + P„B8 >1H $R „BZYjk w+ DQ  $‚ IBZAjk ‚-}Bt‚t$sj $t Q AZ Y‚ A‚R jBb‚„ ABZtI RB Ps„ $R Btjk j$,‚ + jB +  yt‚„‚R$t stI 8sts‚sAj‚ ‚-s8}j‚R R‚‚8 B A‚ t‚‚I‚I qY‚ R8sjj‚R $t‚„‚R$t {sR‚ Q ) 1 € ) BZtIR PB„ Y‚ w€, RDTY‚B„‚8 ; R ) n $t Y‚ VjBtu€j‚$8st w€, RDTY‚B„‚8 bsR RZI$‚I Ak €j‚$8st pk? s„P?sR stI q?BY >xH Y‚$„ jBb‚„ ABZtI $R n stI Z}}‚„ ABZtI dn WB „‚sRBtsAj‚ ABZtIR s„‚ ,tBbt $t ‚t‚„sj stI 8Z{Y „‚8s$tR B A‚ IBt‚ 7$8$js„jk PB„ Y‚ {Bt%‚- js${‚ R‚R s%s$jsAj‚ Z}}‚„ ABZtIR s„‚ }„BAsAjk %‚„k Ps„ P„B8 Y‚ „ZY 3B„ ‚-s8}j‚ bYs $R Y‚ R8sjj‚R + PB„ s Ps8$jk BP {Bt%‚- js${‚ R‚R $t Q b$Y ‚%‚„k Q L d R‚R $t‚„R‚{$t2 3B„ Q ) 1


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