LAUNI NAWA AKE BUƘATA DON FENTIN SARARIN SAMANIYA? GA MA'AUNI NA YAU DA KULLUM, BAI WUCE 2d BA

Ka ɗauki kowace aya ta wani shimfiɗaɗɗen fili ka ba kowacce launi. Doka ɗaya: ayoyi biyu da ke nisan raka’a ɗaya daidai ba za su taɓa samun launi ɗaya ba. Mene ne mafi ƙarancin adadin launuka da ke aiki?

Wannan shi ne matsalar Hadwiger–Nelson, wadda ta samo asali tun 1950 kuma, a cewar marubutan, ɗaya ce daga cikin shahararrun matsaloli da ke buɗe a fannin lissafin siffofi na rarrabe (discrete geometry). Na dogon lokaci an san cewa amsar tana tsakanin 4 da 7. Wani babban ci gaba na kwanan nan ya ɗaga iyakar ƙasa zuwa 5. Har yanzu ba a san amsar daidai ba.

Canza rula

Ba dole ne a auna nisa da rula ta yau da kullum ba. Masana lissafi suna ayyana wasu norms da yawa — hanyoyin auna tsawo — kowacce ana bayyana ta da “ƙwallonta na raka’a” (unit ball), wato saitin ayoyi da ke nisan 1 ko ƙasa da haka daga tsakiya. Ga nisa na yau da kullum ƙwallo ce mai zagaye; ga sauran norms tana iya zama kowace siffa mai lanƙwasa waje (convex) da ke daidaitacciya a kan tsakiyarta.

Ga kowace norm a kan shimfiɗa, amsar wasan fentin tana tsakanin 4 da 7. A girma d, ba ta wuce mai ƙaruwa ta hanyar exponential a d ba ga kowace norm, kuma ga norms na halitta da yawa — har da ta Euclid ta yau da kullum — ita ma aƙalla exponential ce: adadin launuka yana fashewa yayin da girman ke ƙaruwa.

Shin wannan fashewar ita ce ƙa’ida? Noga Alon (Jami’ar Princeton da Jami’ar Tel Aviv), Matija Bucić (Jami’ar Vienna) da James Davies (Jami’ar Leipzig) sun duba norm na yau da kullum (typical). Babu wata hanya ta halitta ta zaɓar norm “ba tare da tsari ba”, don haka sun yi amfani da wani ra’ayi na topology: wani hali yana tabbata ga norm na yau da kullum idan keɓantattun sun zama saiti maras muhimmanci (“meagre”). Aikin baya na Alon, Bucić da Lisa Sauermann ya nuna cewa norm na yau da kullum yana buƙatar launuka 2ᵈ a mafi yawa, kuma ya tambaya yaya kusancin hakan da gaskiya.

Layi ɗaya, ba exponential ba

Amsar: nesa sosai. Sabuwar takardar ta tabbatar da cewa

ga norm na yau da kullum a sararin girma d, launuka 2d koyaushe sun isa;
wannan shi ne mafi kyau da zai yiwu: wani buɗaɗɗen saiti na norms yana buƙatar aƙalla launuka 2d. Don haka wasu norms suna buƙatar daidai 2d.

A girma goma, norm na yau da kullum yana buƙatar launuka ashirin a mafi yawa, yayin da nisa na yau da kullum ke buƙatar adadin da ke ƙaruwa ta hanyar exponential. A cewar marubutan, wannan kuma shi ne karo na farko da aka tantance lambar fenti daidai ga norm “mai lanƙwasa waje sosai” (strictly convex) a kowane girma d.

Iyakar ƙasa tana amfani da wani tarko mai kyau. Nemo ayoyi 2d da duk suke nisan raka’a ɗaya daidai da juna, sai biyu, a da b, waɗanda ke nisan rabin raka’a. Ƙara hoton madubi na dukan tsarin ta hanyar a. Da launuka ƙasa da 2d, duka b da hoton madubinsa za a tilasta musu ɗaukar launin a — amma suna nisan raka’a ɗaya daidai. Saɓani. Wani lemma na kwanciyar hankali yana nuna cewa wannan tsarin yana tsira daga kowane ɗan canji na norm.

Mai gudu shi kaɗai a manyan girma

Iyakar sama tana fentin kowace aya bisa inda wani hasashe (projection) da aka zaɓa da kyau na ta ya faɗi, cikin yanka masu faɗin 1/(2d). Sa ta yi aiki yana buƙatar wani muhimmin sinadari da marubutan suka bayyana a matsayin sigar matrix ta manyan girma na shahararren hasashen mai gudu shi kaɗai (lonely runner conjecture):

sup over x of minᵢ ‖aᵢ · x − bᵢ‖ ≥ k / (2n)

inda ‖t‖ shi ne nisa daga t zuwa cikakkiyar lamba mafi kusa, ga kowane vectors n aᵢ a girma k waɗanda kowane k daga cikinsu ba sa dogara da juna. Wannan bayani kuma ya warware wani hasashe na 1978 na I. J. Schoenberg kan “toshewar gani” (view obstruction) — tambaya game da yadda faɗin yanka masu maimaituwa dole ne ya kasance don toshe kowane gani zuwa iyaka mara ƙarewa — wanda marubutan suka kira ɗaya daga cikin tsofaffin matsaloli da ke buɗe a fannin, da kuma wani hasashe mai alaƙa na Henze da Malikiosis.

Injin cikin godiya

Marubutan sun bayyana a fili: “ChatGPT 6 Pro ya ba mu hujjar sinadari na ƙarshe da muke buƙata a hujjar Theorem 1, wato ta Lemma 7, bayan doguwar tattaunawa”, inda suka raba nasu lura — har da ra’ayin induction da dabarar gaba ɗaya. “An kuma sami hujjar iyakar ƙasa da taimakon ChatGPT 6 Pro.”

Tambayoyi sun rage. An tabbatar da ainihin ƙimar 2d a kan buɗaɗɗen saiti na norms, ba ga duk na yau da kullum ba. Kuma ga nisa na Euclid na yau da kullum, marubutan suna sa ran fiye da launuka 2d a kowane girma — abin da aka riga aka sani a girma 2, 4, 7, 8 da 9 zuwa sama, amma har yanzu a buɗe yake a girma 3, 5 da 6.