Teorema četiri boje — разлика између измена

 
{{short description|Iskazi u matematici}}
[[File:Four Colour Map Example.svg|thumb|primerPrimer četvorobojne mape]]
[[File:Map of United States vivid colors shown.png|thumb|Četvorobojna mapa saveznih država [[Sjedinjene Američke Države|SAD]] (ignorišući jezera).]]
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U [[mathematics|matematici]], '''teorema četiri boje''', ili '''teorema mape s četiri boje''', states that, given any separation of a plane into [[wikt:contiguity|contiguous]] regions, producing a figure called a ''map'', no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet.<ref>From {{harvtxt|Gonthier|2008}}: "Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets. Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map. A point is a corner of a map if and only if it belongs to the closures of at least three regions. Theorem: The regions of any simple planar map can be colored with only four colors, in such a way that any two adjacent regions have different colors."</ref> It was the first major [[theorem]] to be [[computer-assisted proof#List of theorems proved with the help of computer programs|proved using a computer]]. Initially, this proof was not accepted by all mathematicians because the [[computer-assisted proof]] was [[Non-surveyable proof|infeasible for a human to check by hand]].{{sfnp|Swart|1980}} Since then the proof has gained wide acceptance, although some doubters remain.{{sfnp|Wilson|2014|loc=216–222}}
 
U [[mathematics|matematici]], '''teorema četiri boje''', ili '''teorema mape ssa četiri boje''', statesnavodi that,da givenza anybilo separationkoje ofrazdvajanje aravni planena intosusedne [[wikt:contiguity|contiguous]] regionsregione, producingčime ase figureformira calledslika akoji se naziva ''mapmapa'', nonije morepotrebno thanviše fourod colorsčetiri areboje requiredda tobi colorse theobojili regionsregioni ofkarte thetako mapda sonijedan thatpar nosusednih tworegiona adjacent regions have thenema sameistu colorboju. ''AdjacentSusedni'' meansznači thatda twodva regionsregiona sharedele azajednički commonsegment boundarygranične curve segmentkrive, nota merelyne asamo cornerugao wheregde threese orsusreću moretri regionsili više meetregiona.<ref>From {{harvtxt|Gonthier|2008}}: "Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets. Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map. A point is a corner of a map if and only if it belongs to the closures of at least three regions. Theorem: The regions of any simple planar map can be colored with only four colors, in such a way that any two adjacent regions have different colors."</ref> ItOvo wasje thebila firstprva majorznačajna [[theoremteorema]] tokoja beje [[computer-assisted proof#List of theorems proved with the help of computer programs|proveddokazana]] usingpomoću a computer]]računara. Initially,U thispočetku proofovaj wasdokaz notnisu acceptedprihvatili bysvi allmatematičari, mathematiciansjer becauseje thebilo nemoguće da se [[computerNon-assistedsurveyable proof|manuelno]] wasproveri [[NonComputer-surveyableassisted proof|infeasible for a human to check bykompjuterski handdokaz]].{{sfnp|Swart|1980}} SinceOd thentada theje proofdokaz hasstekao gainedširoko wide acceptanceprihvatanje, althoughmada ima onih koji i dalje someosporavaju doubtersnjegovu remainvalidnost.{{sfnp|Wilson|2014|loc=216–222}}
The four color theorem was proved in 1976 by [[Kenneth Appel]] and [[Wolfgang Haken]] after many false proofs and counterexamples (unlike the [[five color theorem]], a theorem that states that five colors are enough to color a map, which was proved in the 1800s). To dispel any remaining doubts about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally, in 2005, the theorem was proved by [[Georges Gonthier]] with general-purpose [[proof assistant|theorem-proving software]].
 
Teoremu četiri boje su dokazali [[Kenneth Appel|Kenet Apl]] i [[Wolfgang Haken|Volfgang Hejken]] 1976. godine, nakon mnogih lažnih dokaza i protivprimera (za razliku od [[five color theorem|teoreme pet boja]], teoreme koja navodi da je pet boja dovoljno za bojenje mape, što je dokazano 1800-ih). Da bi se razvejale sve preostale nedoumice oko dokaza Apel-Hejkena, jednostavniji dokaz koji koristi iste ideje i koji se još uvek oslanja na računare objavili su Robertson, Sanders, Sejmour i Tomas 1997. godine. Pored toga, 2005. godine teoremu je dokazao [[Georges Gonthier|Žorž Gontje]], softverom opšte namene za [[proof assistant|dokazivanja teorema]].
 
== Precizna formulacija teoreme ==