. ▁The ▁graph ▁should ▁be ▁drawn ▁in ▁the ▁plane ▁with ▁each ▁vertex ▁as ▁a ▁point , ▁each ▁edge ▁as ▁a ▁curve ▁connecting ▁its ▁two ▁end points , ▁and ▁no ▁vertex ▁placed ▁on ▁an ▁edge ▁that ▁it ▁is ▁not ▁incident ▁to . ▁A ▁crossing ▁is ▁counted ▁whenever ▁two ▁edges ▁that ▁are ▁dis joint ▁in ▁the ▁graph ▁have ▁a ▁non empty ▁intersection ▁in ▁the ▁plane . ▁The ▁question ▁is ▁then , ▁what ▁is ▁the ▁minimum ▁number ▁of ▁cross ings ▁in ▁such ▁a ▁drawing ? ▁ ▁Tur án ' s ▁form ulation ▁of ▁this ▁problem ▁is ▁often ▁recognized ▁as ▁one ▁of ▁the ▁first ▁studies ▁of ▁the ▁crossing ▁numbers ▁of ▁graphs . ▁( An other ▁independent ▁form ulation ▁of ▁the ▁same ▁concept ▁occurred ▁in ▁soci ology , ▁in ▁methods ▁for ▁drawing ▁soci og rams , ▁and ▁a ▁much ▁older ▁puzz le , ▁the ▁three ▁util ities ▁problem , ▁can ▁be ▁seen ▁as ▁a ▁special ▁case ▁of ▁the ▁brick ▁factory ▁problem ▁with ▁three ▁kil ns ▁and ▁three ▁storage ▁facilities .) ▁Cross ing ▁numbers ▁have ▁since ▁gained ▁greater ▁importance , ▁as ▁a ▁central ▁object ▁of ▁study ▁in ▁graph ▁drawing ▁and ▁as ▁an ▁important ▁tool ▁in ▁V L SI ▁design ▁and ▁discrete ▁geometry . ▁ ▁Upper ▁bound ▁Both ▁Zar ank iew icz ▁and ▁Kaz im ierz ▁Urban ik ▁saw ▁Tur án ▁speak ▁about ▁the ▁brick ▁factory ▁problem ▁in ▁different ▁tal ks ▁in ▁Poland ▁in ▁ 1 9 5 2 , ▁and ▁independently ▁published ▁attempted ▁solutions ▁of ▁the ▁problem , ▁with ▁equivalent ▁formulas ▁for ▁the ▁number ▁of ▁cross ings . ▁As