A set S of integers is said to be sum-free if
a
,
b
∈
S
a,b \in S
implies
a
+
b
∉
S
a + b \notin S
. In this paper, we investigate two new problems on sum-free sets: (1) Let
f
(
k
)
f(k)
denote the largest positive integer for which there exists a partition of
{
1
,
2
,
…
,
f
(
k
)
}
\{ 1,2, \ldots ,f(k)\}
into k sum-free sets, and let
h
(
k
)
h(k)
denote the largest positive integer for which there exists a partition of
{
1
,
2
,
…
,
h
(
k
)
}
\{ 1,2, \ldots ,h(k)\}
into k sets which are sum-free
mod
h
(
k
)
+
1
\bmod h(k) + 1
. We obtain evidence to support the conjecture that
f
(
k
)
=
h
(
k
)
f(k) = h(k)
for all k. (2) Let
g
(
n
,
k
)
g(n,k)
denote the cardinality of a largest subset of
{
1
,
2
,
…
,
n
}
\{ 1,2, \ldots ,n\}
that can be partitioned into k sum-free sets. We obtain upper and lower bounds for
g
(
n
,
k
)
g(n,k)
. We also show that
g
(
n
,
1
)
=
[
(
n
+
1
)
/
2
]
g(n,1) = [(n + 1)/2]
and indicate how one may show that for all
n
⩽
54
,
g
(
n
,
2
)
=
n
−
[
n
/
5
]
n \leqslant 54,g(n,2) = n - [n/5]
.