Intersecting and CrossIntersecting Families of Labeled Sets A family cal A of sets is said to be  intersecting  if any two sets in cal A intersect. Families cal A_1, ..., cal A_p are said to be  crossintersecting  if, for any i, j in 1, ..., p such that i neq j, any set in cal A_i intersects any set in cal A_j.   For bf k  (k_1, ..., k_n) in Bbb Nn, 2 leq k_1 leq ... leq k_n, let cal L_bfk be the family of  labeled nsets  given by cal L_bfk : (1,l_1), ..., (n,l_n) colon l_i in 1, ..., k_i, i  1, ..., n. We point out a relationship between intersecting families and crossintersecting families of labeled sets, and we show that, if cal A_1, ..., cal A_p are crossintersecting subfamilies of cal L_bfk, then  sum_j  1p cal A_j leq left  matrix k_1k_2...k_n u0026 hboxif p leq k_1;cr pk_2...k_n u0026 hboxif p geq k_1.cr  right.   We also determine the cases of equality.  We then obtain a more general inequality, a special case of which is a sharp bound for crossintersecting families of  permutations .

Shifted Set Families, Degree Sequences, and Plethysm We study, in three parts, degree sequences of kfamilies  (or kuniform hypergraphs) and shifted kfamilies.    bullet  The first part collects for the first time in one place, various implications  such as   scriptstyle hboxThreshold Rightarrow  hboxUniquely Realizable Rightarrow hboxDegreeMaximal Rightarrow hboxShifted   which are equivalent concepts for 2families ( simple graphs), but  strict implications for kfamilies with k geq 3.  The implication that uniquely realizable implies degreemaximal seems to be new.  bullet The second part recalls Merris and Robyu0027s reformulation of the characterization due to Ruch and Gutman for graphical degree sequences and shifted 2families. It then introduces two generalizations which are characterizations of shifted kfamilies.  bullet  The third part recalls the connection between degree sequences of kfamilies of size m and the plethysm of elementary symmetric functions e_me_k. It then uses highest weight theory to explain how shifted kfamilies provide the top part of these plethysm expansions, along with offering a conjecture about a further relation.