Extremal t -intersecting sub-families of hereditary families

Erdős–Ko–Rado theorems for simplicial complexes

“…The following lemma follows from Borg's more general result [5,Theorem 2.7]. We use algebraic shifting to give a short new proof of the specific result.…”

“…By Lemma 2.11 and the following discussion, the minimum facet cardinality of Shift F is depth F To the best of my knowledge, Theorem 3.3 is the first 'new' intersection theorem to be proved by algebraic shifting. property for any r-family, hence a reduction to [5,Theorem 2.7] similar to that in Theorem 3.3 will show that if is a t-fold near-cone (i.e., shifted with respect to its first t elements) with depth equal to its minimum facet dimension, then [5, Conjecture 2.7] holds for uniform r-families of faces in . In particular, [5,Conjecture 2.7] holds for sequentially Cohen-Macaulay t-fold near-cones.…”

“…The natural extension was made by Borg [5]: [5,Conjecture 1.7].) If is a simplicial complex having minimal facet cardinality k, then is r-EKR for r k 2 .…”

“…We refer the reader to [6] for additional background on the r-EKR property in graphs, and to [5] for further relationships with more general intersection problems. This paper is organized as follows.…”

An Analogue of the Hilton-Milner Theorem for Weak Compositions

“…Some recent work done on this problem and its variants can be found in [5,7,8,11,12,19,26,33,34,35,36,42]. The investigation of the Erdős-Ko-Rado property for graphs started in [23], and gave rise to [4,6,21,22,24,44]. The Erdős-Ko-Rado type results also appear in vector spaces [9,18], set partitions [27,28,31] and weak compositions [30,32].…”