Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Further Results on Existentially Closed Graphs Arising from Block
  Designs   A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets
$A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup
B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design
with block set $\cal B$, its block intersection graph is the graph whose vertex
set is $\cal B$ and two vertices (blocks) are adjacent if they have non-empty
intersection.
  In this paper, we investigate the block intersection graphs of pairwise
balanced designs, and propose a sufficient condition for such graphs to be
$2$-e.c. In particular, we study the $\lambda$-fold triple systems with
$\lambda \ge 2$ and determine for which parameters their block intersection
graphs are $1$- or $2$-e.c. Moreover, for Steiner quadruple systems, the block
intersection graphs and their analogue called $\{1\}$-block intersection graphs
are investigated, and the necessary and sufficient conditions for such graphs
to be $2$-e.c. are established.

16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph