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COOLMATH!
CURIOUSMATHEMATICSFORFUNANDJOYAPRIL2016PROMOTIONALCORNER:Haveyouanevent,aworkshop,awebsite,somematerialsyouwouldliketosharewiththeworldLetmeknow!
Iftheworkisaboutdeep,joyous,andrealmathematicaldoingI'llhappilymentionithere.
***Peopledomathvideos!
CheckoutMarcChamberlain'shttps://www.
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WhydoesworkTHISMONTH'SPUZZLER:Itispossibletocolorthefirsteightcountingnumberseacheitherredorbluesothatweneverhavethreedistinctintegers,,andallthesamecolor.
CanthesametaskbecompletedwiththefirstninecountingnumbersWhatisthesmallestsothateverycoloringofthenumbers1,2,3,…,eitherredorblueissuretohaveamonochromatictripleHowdoestheanswerchangeifwepermitgeneric"triples"with(Nowweneedeachandtobedistinctcolorstoo.
)JamesTanton2016www.
jamestanton.
comandwww.
gdaymath.
comRAMSEYTHEORYHereisaclassicresult:Ifsixuniversitystudentsareselectedatrandom,thenthereissuretobeeitherthreestudentsamongthesixwhoaremutualfriendsorthreestudentswhoaremutualstrangers(orboth).
(Weareassumingherethatfriendshipisreciprocal:IfAlbertisfriendswithBilbert,thenBilbertisalsofriendswithAlbert.
Beingastrangerisreciprocaltoo.
)Here'sthereasoning:Chooseoneofthesixstudents,Cuthbert.
Therearefiveotherstudentseachofwhichheiseitherfriendswithorastrangerto.
SupposeCuthbertisfriendswithamajorityofthesefive,thatis,friendswithatleastthreeofthem.
(If,instead,heisastrangertoamajority,thenswitchthewordsfriendandstrangerinwhatfollows.
)Amongthesethreepeople,ifanytwoaremutualfriends,thenwehaveatripleoffriends:Cuthbertandthosetwo.
Ifnoneofthosethreearefriends,thenwehavefoundatripleofstrangers.
Theresultisnottrueforjustfivepeopleselectedatrandomasseenbythisgraphic.
Hereeachdotrepresentsastudentandarededgeindicatesmutualfriendsandablueedgemutualstrangers.
Nothreepeopleareconnectedbyedgesallofthesameonecolor.
Intermsofcoloreddiagrams,ourpartyresulttranslatesasfollows:Drawsixdotsonapageandthe15edgesbetweenallpossiblepairsofdots.
Itisimpossibletocolorthoseedgesredandblueandavoidamonochromatictriangle.
Togeneralizethisidealetdenotetheleastnumberofdotsoneneedstodrawonapagesothatifweconnectallpairsofdotswitheitherredorblueedges,thereissuretobeeitherasetofdotswithalltheedgesamongthemredorasetofdotswithalltheedgesamongthosedotsblue.
(Thisisassumingthatsuchaleastnumberexists!
Maybenomatterhowmanydotsonedrawsonecanalwaysavoidred"cliques"ofsizeandbluecliquesofsize)Theideaofstudyingthenecessarysizeofasystemtoensurecertainsub-substructuresexistswasfirstformallyexploredbyBritishmathematicianFrankRamsey(1903–1930).
ThisworkistodaycalledRamseyTheoryinhishonor.
Ourpartyresultreadsas.
(Drawsixdotsandcolortheedgesbetweenthenredandblue.
Eitheraredtriangleissuretoappearorablueone.
)Itisnothardtoseethat.
(Ifwedrawdotsonapageandcolortheedges,theneitheroneisredandwe'vefoundredcliqueofsizeoralledgesareblueandwehaveabluecliqueofsize.
Also,isnotorsmaller:coloringalltheedgesbetweendotsblueillustratesthis.
)JamesTanton2016www.
jamestanton.
comandwww.
gdaymath.
comComputingRamseynumbersisstillaveryactiveareaofresearch.
Onlythesefewvaluesarecurrentlyknown.
(Ofcourse,:justswitchcolors.
)Generalizing…Setastheleastnumberofdotsoneneedstodrawonthepagetoensurethat,incoloringtheedgesred,blueandgold,eitheracliqueofdotswithnothingbutrededgesbetweenthem,oracliqueofdotswithnothingbutblueedgesbetweenthem,oracliqueofdotswithnothingbutgoldedgesbetweenthemissuretoappear.
Itisknownthat.
(Draw17dotsonapageandcoloreachofthe153edgesbetweenthemeitherred,blue,orgold.
Thenamonochromatictriangleissuretoappear.
Also,itispossibletoavoidmonochromatictriangleswithonly16dotsonthepage.
)Andforfullgeneralitysetastheleastnumberofdotsoneneedstodrawonapagesothat,incoloringeachoftheedgesbetweenapairofdotsoneofcolors,thereissuretobeacliqueofdotswithalltheedgesbetweenthemthethcolor,forsome.
Ofcourse,weareassumingthatthisnumberexists-thatthereisaleastnumberofdotsthatassuresamonochromaticstructureappears.
Ramsey'sTheorem:Eachisindeedameaningfulfinitenumber.
Let'sillustratewhy.
ThevaluedoesnotappearonthelistofknownRamseynumbers.
Butwecanprovethatitisafinitenumber.
Wehave,fromthelist,and.
Drawdotsonthepageandcolortheedgesbetweenthemredandblue.
Weshallnowreasonthateitheracliqueofdotsexistswithalledgesbetweenthemredoracliqueofdotsexistswithalledgesbetweenthemblue.
Thiswillestablishthat.
Inourdiagramofdotswithedgescolored,chooseoneparticulardot.
CallitDilbert.
Dilberthassomerededgesemanatingfromitconnectingitto,say,otherdots.
TheremainingedgesemanatingfromDilbertareblue,connectingtootherdots,say.
Here.
Nowitcan'tbethatbothand.
Soeitherisatleastorisatleast.
Case:ConsiderthedotsthatconnecttoDilbertbyrededges.
Becausethereiseitheraredcliqueofamongthesedotsorthereisbluecliqueofamongthem.
Ifthereisaredcliqueof3,thenincludingDilbertintheclique(alledgestoDilbertarered)actuallymeanswehavearedcliqueof,oneofthetwostructureswearehopingtoseefor.
If,ontheotherhand,thereisabluecliqueof,thenwehaveabluecliqueof!
Eitherwaywehavefoundoneofthetwothingswearelookingfor.
JamesTanton2016www.
jamestanton.
comandwww.
gdaymath.
comCase:ConsiderthedotsthatconnecttoDilbertviablueedges.
Because,amongthesedotsthereiseitheraredcliqueof(oneofthepossibilitieswewerehopingfor)orabluecliqueof.
Inthelattercase,sincealltheedgestoDilberthereareblue,addingDilberttothecliqueoffiveactuallymakesabluecliqueof!
Again,wearesuretohaveatleastoneofthetwostructureswewerelookingfor.
Ingeneral,onecanprovejustthiswaytheinequality:.
ThenfromknowingthatRamseynumberswithsmallerindicesarefinitewecanreasonthateveryRamseynumberisfinite.
GeneralizedRamsey'sTheorem:Eachvalueisfinite.
Wehavejustshownthateachofthevaluesfortwocoloringsisafinitenumber.
Let'sshowhowwecanusethisfacttoestablishthateachofthenumbersforthreecoloringsmustalsobefinite.
Consider.
Wewanttoshowthatthereisanumbersothatifwedrawdotsonthepageandcolortheedgeseitherred,blue,orgold,thereissuretobeeitheraredcliqueofdots,orabluecliqueofdots,oragoldcliqueofdots.
Sometimeswhenwesquintoureyes,redandbluecanstarttoeachlookpurple.
Soadiagramwithedgespaintedwiththreecolors,red,blue,andgold,canlooklikeadiagramwithedgespaintedjusttwocolors,purpleandgold,undersquintyeyes.
Thisgivesawaytobringthree-coloringsbacktotwo-colorings.
Let.
(Soanydiagramofdotswithedgespaintedredandbluehaseitheraredcliqueofdotsorabluecliqueofdots.
)Let.
(Soanydiagramofdotswithedgespaintedpurpleorgoldhaseitherapurplecliqueofdotsoragoldcliqueofdots.
)Nowdrawdotsonthepageandcolortheedgesred,blue,andgold.
(Remember,wearelookingforeitheraredcliqueofdotsorabluecliqueofdotsoragoldcliqueofdots.
)Squintyoureyesandseeonlypurpleandgold.
Byourchoiceofwe'reeitherseeingapurplecliqueofdotsoragoldcliqueofdots.
Ifwe'reinthelattercase,thenwe'vefoundoneofthethreethingswewerehopingtosee.
Ifwe'reintheformercase,thenweareseeingapurplecliqueofdots,which,whenweunsquintoureyes,isasetofdotswithredandblueedgesbetweenthem.
Butourchoiceofwasspecial:itguaranteesthateitherwehavearedcliqueofdotsorabluecliqueofdots.
Soagain,weareseeingoneofthethreethingswewerehopingtosee.
Soisfiniteanumber:itisboundedbythenumberwith.
Ingeneral,onereasonsthiswaytoshowthatwith.
Nowknowingthatallthethree-colorRamseyvaluesarefinite,weJamesTanton2016www.
jamestanton.
comandwww.
gdaymath.
comcanusethistoarguethatallthefour-colorRamseynumbersarefinite,whichleadstoallthefive-colorRamseynumbersbeingfinite,andsoon.
CONNECTIONSTOTHEOPENINGPUZZLERHere'saboldclaim:Itisimpossibletocolorthecountingnumberseachoneoffiftypossiblecolorsandavoidamonochromatictriple,,.
(Thegenericcaseisallowed.
)(Thenumber50isimmaterialhere:anyfinitenumberofcolorswilldo!
)Here'swhy.
WejustprovedthattheRamseyvalue,withfiftycolors,isafinitevalue.
Letbeitsvalue.
Soifwedrawdotsonapageandcolortheedgesusingfiftydifferentcolors,thenwearesuretofindamonochromaticcliqueofthree.
Thatis,we'dfindamonochromatictriangle.
Supposewehavecoloredthecountingnumbers1,2,3,…eachoneoffiftycolors.
Drawadotaboveeachofthefirstcountingnumbersanddrawanedgebetweeneachpairdots.
Nowcoloreachedgeaccordingtothefollowingrule:Painttheedgeconnectingthenumbertothenumber(assumehere)withthecolorofnumber.
Amonochromatictriangleissuretoexist.
Fromthistrianglewehavethatthecolorofisthesamethecolorof,whichisthesameasthecolorof.
Butobserve:.
Wehavefoundthreenumbers,,andallthesamecolor.
Exercise:Coloreachpositiveintegeronecolorfromagivenfinitesetofcolors.
Musttherebeamonochromatictriple,,RESEARCHCORNER1.
Letbethesmallestvaluesothatifwecolortheeachofthenumberswithoneofcolorsthereissuretobeamonochromatic"triple".
(Wejustprovedthatexistsand,byeasyextension,thateachvalueexists.
)Wehaveand(ifyoudidthesecondpartoftheopeningexercise).
Canyoudetermineanyothervaluesof2.
Letbethesmallestvaluesothatifwecolortheeachofthenumberswithoneofcolorsthereissuretobeamonochromatictriple.
Wehaveand.
CanyouadjustthepreviousprooftoestablishthatthevaluesexistJamesTanton2016www.
jamestanton.
comandwww.
gdaymath.
com3.
Explorecoloringthepositiveintegerswithafinitepaletteofcolorsandestablishingtheexistenceofamonochromaticquadruple,,,,with.
2016JamesTantontanton.
math@gmail.
com