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DESCRIPTION: The  n -cube is the poset obtained by ordering all subsets of 
 {1\,2\,...\, n } by inclusion. A symmetric chain is a sequence of subsets  
 A k  ⊆ A   k +1 ⊆…⊆ A n-k   with | A i  |= i  for all  i=k \,…\, n-k \, and
  a symmetric chain decomposition\, or SCD for short\, of the  n -cube is a 
 partition of all its elements into symmetric chains. There are several know
 n descriptions of SCDs in the  n -cube for any  n ≥1\, going back to works 
 by De Bruijn\, Aigner\, Kleitman and several others. All those construction
 s\, however\, yield the very same SCD.   In this talk I will present severa
 l new constructions of SCDs in the  n -cube. Specifically\, we construct fi
 ve pairwise edge-disjoint SCDs in the  n -cube for all  n ≥90\, and four pa
 irwise orthogonal SCDs for all  n ≥60\, where orthogonality is a slightly s
 tronger requirement than edge-disjointness. Specifically\, two SCDs are cal
 led orthogonal if any two chains intersect in at most a single element\, ex
 cept the two longest chains\, which may only intersect in the unique minima
 l and maximal element (the empty set and the full set). This improves the p
 revious best lower bound of three orthogonal SCDs due to Spink\, and is ano
 ther step towards an old problem of Shearer and Kleitman from the 1970s\, w
 ho conjectured that the  n -cube has ⌊ n /2⌋+1 pairwise orthogonal SCDs.   
 We also use our constructions to prove some new results on the central leve
 ls problem\, a far-ranging generalization of the well-known middle two leve
 ls conjecture (now theorem)\, on Hamilton cycles in subgraphs of the (2 n +
 1)-cube induced by an even number of levels around the middle. Specifically
 \, we prove that there is a Hamilton cycle through the middle four levels o
 f the (2 n +1)-cube\, and a cycle factor through any even number of levels 
 around the middle of the (2 n +1)-cube.   This talk is based on two papers\
 , jointly with Sven Jäger\, Petr Gregor\, Joe Sawada\, and Kaja Wille (ICAL
 P 2018)\, and with Karl Däubel\, Sven Jäger\, and Manfred Scheucher\, respe
 ctively. 
DTSTAMP:20190115T033900
DTSTART:20190121T160000
CLASS:PUBLIC
LOCATION:Technische Universität Berlin\n Institut für Mathematik\n Straße d
 es 17. Juni 136\n 10623 Berlin\n Room MA 041 (Ground Floor)
SEQUENCE:0
SUMMARY:Torsten Mütze (Technische Universität Berlin): On symmetric chains 
 and Hamilton cycles
UID:95604715@/www.mi.fu-berlin.de
URL:https://www.mi.fu-berlin.de/en/facetsofcomplexity/monday/20190121-C-Mue
 tze.html
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