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.

Extremal Problems for tPartite and tColorable Hypergraphs Fix integers t ge r ge 2 and an runiform hypergraph F. We prove that the maximum number of edges in a tpartite runiform hypergraph on n vertices that contains no copy of F is c_t, Fn choose r o(nr), where c_t, F can be determined by a finite computation.   We explicitly define a sequence F_1, F_2, ldots of runiform hypergraphs, and prove that the maximum number of edges in a tchromatic runiform hypergraph on n vertices containing no copy of F_i is alpha_t,r,in choose r o(nr), where alpha_t,r,i can be determined by a finite computation for each ige 1. In several cases, alpha_t,r,i is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.