- Series
- Combinatorics Seminar
- Time
- Friday, September 17, 2010 - 3:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Jerry Griggs, Carolina Distinguished Professor and Chair – Mathematics, University of South Carolina
- Organizer
- Prasad Tetali
Given a finite poset P, we consider the largest size La(n,P) of a family of subsets of [n]:={1,…,n} that contains no subposet HP.Sperner′sTheorem(1928)givesthat{\rm La}(n,P_2)= {n\choose{\lfloor n/2\rfloor}},whereP_2isthetwo−elementchain.Thisproblemhasbeenstudiedintensivelyinrecentyears,anditisconjecturedthat\pi(P):= \lim_{n\rightarrow\infty} {\rm La}(n,P)/{n\choose{\lfloor n/2\rfloor}}existsforgeneralposetsP,and,moreover,itisaninteger.Fork\ge2letD_kdenotethek−diamondposet\{A< B_1,\ldots,B_k < C\}.WestudytheaveragenumberoftimesarandomfullchainmeetsaP−freefamily,calledtheLubellfunction,anduseitforP=D_ktodetermine\pi(D_k)forinfinitelymanyvaluesk.Astubbornopenproblemistoshowthat\pi(D_2)=2;hereweprove\pi(D_2)<2.273$ (if it exists). This is joint work with Wei-Tian Li and Linyuan Lu of University of South Carolina.