Almost regular subhypergraph of hypergraph with large minimal degree

I am interested in knowing whether the following conjecture is true or not:

For every $d \geq 1$ there exist constants $\delta,M_0 > 0$ such that the following holds for all $M \geq M_0$.

Let $\mathcal{S}$ be a finite family of sets of size at least $M$ in which each element appear at most $d$ times.

There exists a set $X$ such that $\delta M \leq |S \cap X| \leq M$ for all $S \in \mathcal{S}$.

We can also formulate this in terms of hypergraphs:

For every $d \geq 1$ there exist constants $\delta,M_0 > 0$ such that the following holds for all $M \geq M_0$.

Let $H$ be a $d$-uniform hypergraph in which each vertex has degree at least $M$. There is a subhypergraph $H'$ of $H$, obtained by only deleting hyperedges, in which the degree of each vertex is between $\delta M$ and $M$.

(If we want to make the statements completely equivalent, we should ask all hyperedges to have uniformity at most $d$ rather than exactly $d$.)

The question has the flavor of Hall's matching theorem and of Beck-Fiala, but so far I was unable to prove the conjecture using these tools. Perhaps I am missing some simple argument, or perhaps there is a simple counterexample.

Asked by user40 (1 )
Aug 01, 2017 at 16:24