Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On set systems without a simplex-cluster and the Junta method   A family $\{A_{0},\ldots,A_{d}\}$ of $k$-element subsets of
$[n]=\{1,2,\ldots,n\}$ is called a simplex-cluster if $A_{0}\cap\cdots\cap
A_{d}=\varnothing$, $|A_{0}\cup\cdots\cup A_{d}|\le2k$, and the intersection of
any $d$ of the sets in $\{A_{0},\ldots,A_{d}\}$ is nonempty. In 2006, Keevash
and Mubayi conjectured that for any $d+1\le k\le\frac{d}{d+1}n$, the largest
family of $k$-element subsets of $[n]$ that does not contain a simplex-cluster
is the family of all $k$-subsets that contain a given element. We prove the
conjecture for all $k\ge\zeta n$ for an arbitrarily small $\zeta>0$, provided
that $n\ge n_{0}(\zeta,d)$.
  We call a family $\{A_{0},\ldots,A_{d}\}$ of $k$-element subsets of $[n]$ a
$(d,k,s)$-cluster if $A_{0}\cap\cdots\cap A_{d}=\varnothing$ and
$|A_{0}\cup\cdots\cup A_{d}|\le s$. We also show that for any $\zeta n\le
k\le\frac{d}{d+1}n$ the largest family of $k$-element subsets of $[n]$ that
does not contain a $(d,k,(\frac{d+1}{d}+\zeta)k)$-cluster is again the family
of all $k$-subsets that contain a given element, provided that $n\ge
n_{0}(\zeta,d)$.
  Our proof is based on the junta method for extremal combinatorics initiated
by Dinur and Friedgut and further developed by Ellis, Keller, and the author.

6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex