The [Erdős–Gallai theorem](http://en.wikipedia.org/wiki/Erd%25C5%2591s%25E2%2580%2593Gallai_theorem) gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph.


In particular $d\_1 \ge d\_2 \ge \dots \ge d\_n$ is the degree sequence of a graph on $n$ vertices if and only if


(1) $d\_1 + d\_2 + \dots d\_n$ is even, and


(2) $$ \sum\_{i=1}^k d\_i \le k(k-1) + \sum\_{i=k+1}^n\min (d\_i, k) $$


holds for $1 \le k \le n$.


Now let $\Delta$ be a $k$-dimensional simplicial complex on $n$ vertices, with a complete $(k-1)$-skeleton. I.e. $$f\_{k-1} (\Delta) = {n \choose k} $$


The degree of a $(k-1)$-dimensional face of $\Delta$ is defined to be the number of $k$-dimensional faces containing it.


What are the possible degree sequences
$$d\_1 \ge d\_2 \ge \dots \ge d\_{n \choose k}?$$


Clearly a necessary condition is that $(k+1)$ divides the sum $d\_1 + d\_2 + \dots$, but is there something analogous to condition (2) above that makes this into necessary and sufficient conditions?


Is the [Kruskal-Katona theorem](http://en.wikipedia.org/wiki/Kruskal%25E2%2580%2593Katona_theorem) of any help? 



Is there an analogue of the Erdős–Gallai theorem for simplicial complexes?
Very little is known about the question (and even about the easier case of vertex degrees), and it contains as a special case some notoriously hard questions: For example the case that all $d\_i$s are equal to 1 (or to some $\lambda$ is the question on the existence of combinatorial designs of certain parameters. I suppose that even checking if a sequence is a dgree sequence is computationally intractable, for large values of $d$ but I am not sure about that. 


