Referee's Report on Manuscript #1358: 
``Families of sets with intersecting clusters"

Authors : Chen, Liu and Wang


Comments:

This is a newer, nicer, revised version of a submission from more than a year ago.
The authors tackle a conjecture of Mubayi on intersecting families from Ref.[7]. 
The main technical contribution consists of proving Mubayi's conjecture in a
special case (Theorem 2.1), proving a similar (to the conjectured)  result under a modified hypothesis 
(Theorem 3.1), and finally proving a slightly weaker 
version (Theorem 4.4) of the conjecture  for large n, the size of the ground set. 
In the following, the statements marked by (*) contain original contribution
by the authors, beyond the work of Mubayi [7,8].

(*) The proof of Theorem 2.1 is short and is based on a useful observation.

While the result in Theorem 3.1 is perhaps impressive, the proof follows very closely
that of a method used by Mubayi in [7]. 

(*) The main difference is by way of Lemma 3.3, which is an extension of Lemma 3 of [7], and differs in proof from that in  [7].

The main addition to this REVISED version is Theorem 4.4. Once again the proof 
follows the other work of Mubayi, namely  [8].  Perhaps more unfortunately, 
there is another recent submission to the same journal
(by different authors) seemingly proving  Theorem 4.4, under
a weaker hypothesis. Namely, this other work proves the Mubayi conjecture (for large $n$), in its original form, 
without the additive term (d-4)/2 in the hypothesis on the size of the union of the intersection-free sets.

Recommendation and Suggestions:

Due to the prestige of the journal JCTA,
the referee can not recommend publication. 

The manuscript is written clearly and is easy to read. Theorems 2.1 and 3.1
deserve publication, and  the referee encourages submission to an alternate 
(perhaps less competitive) journal such as Graphs & Combinatorics or Discrete Mathematics.
Finally, the referee feels badly for the authors, in light of independent 
developments (mentioned above).