TITLE: Erdos Ko Rado for hypergraphs of bounded degree?
QUESTION [5 upvotes]: The Erdos-Ko-Rado theorem states that if $H$ is a $k$-uniform hypergraph on $[n]$ which is intersecting, then $|H| \leqslant  \binom{n-1}{k-1}$. The easy example which shows this is tight is just take $H$ to be all $k$-subsets containing the element $1$. This example however the degree of the vertex $1$ would be $|H|$. I'm interested if the bound can be improved if we assume bounds on the max degree of $H$, in particular if the max degree is on the order of the average degree. Does anyone know of such a theorem?

REPLY [2 votes]: I returned to this problem after awhile and I can get a slight improvement and show a nice connection between simple graphs and multigraphs. Throughout the following, I will assume that the uniformity, $k$, is fixed and the max degree, $\Delta$ is tending to infinity. 
Now, let $g(k,\Delta)$ denote the maximimum number of edges in a simple intersecting $k$-graph with max degree $\Delta$, and let $g^*(k,\Delta)$ denote the equivalent number for multigraphs. In my previous post, I argued three things:

$g^*(k-1,\Delta)\leq g(k,\Delta)\leq g^*(k,\Delta)$
$g^*(k,\Delta)\leq k\Delta$
$g^*(k,\Delta)\geq (k-1+{1\over k})\Delta$ for infinitely many $k$.

I can show an improvement to item 1. when $k$ is fixed and $\Delta\to\infty$ which gives an answer of $g(k,\Delta)\leq (k-1)\Delta+C_k$ where $C_k$ is some (large) constant depending only on $k$.
In fact, I'll show that $g(k,\Delta)\leq g^*(k-1,\Delta)+C_k$. To do so, let $G$ be a simple intersecting $k$-graph with max degree $\Delta$ and set $C_k=k!\cdot k^k$. The value of $C_k$ comes from the Erdos-Rado sunflower lemma. In particular, if $|E(G)|>C_k$, then $G$ contains a sunflower with $k+1$ petals. Since this is the case, we can pull off sunflowers $S_1,\dots,S_\ell$ from $G$, each having $k+1$ petals, where $\ell$ is the smallest integer with $|E(G)|\leq\ell(k+1)+ C_k$..
Now, let $S_i'$ denote the core of $S_i$. Since $G$ was a simple graph, each $S_i'$ has size at most $k-1$. Thus, if $G'$ is the multigraph with edge set $S_i'$, then $G'$ is $\leq(k-1)$-uniform. I claim that $G'$ is actually intersecting as well. Indeed, this follows from the fact that each $S_i$ is a $k$-uniform sunflower with $k+1$ petals, so since $G$ was intersecting, it must be the case that the cores of each of these sunflowers intersect as well.
Finally, if $\Delta'$ is the max degree of $G'$, we have $\Delta'\leq \Delta/(k+1)$ since each core corresponds to $k+1$ edges of $G$. Hence, $\ell=|E(G')|\leq g^*(k-1,\Delta/(k+1))$. It is not difficult to observe that $t\cdot g^*(s,\Delta)\leq g^*(s,t\Delta)$ by stacking multiple copies of a graph on itself, so we finally have
$$
|E(G)|\leq \ell(k+1)+C_k\leq (k+1)\cdot g^*(k-1,\Delta/(k+1))+C_k\leq g^*(k-1,\Delta)+C_k.
$$
Thus, $g(k,\Delta)\leq g^*(k-1,\Delta)+C_k$. This immediately implies that $g(k,\Delta)\leq (k-1)\Delta+C_k$, which is better than the trivial bound of $k\Delta$ for large $\Delta$.
Unfortunately, I do not know any way to improve the trivial bound of $g^*(k,\Delta)\leq k\Delta$.