diff --git a/src/backend/utils/mmgr/aset.c b/src/backend/utils/mmgr/aset.c
index 71a396d4456..654d16bc8e8 100644
--- a/src/backend/utils/mmgr/aset.c
+++ b/src/backend/utils/mmgr/aset.c
@@ -89,7 +89,8 @@
  *
  * With the current parameters, request sizes up to 8K are treated as chunks,
  * larger requests go into dedicated blocks.  Change ALLOCSET_NUM_FREELISTS
- * to adjust the boundary point.
+ * to adjust the boundary point; and adjust ALLOCSET_SEPARATE_THRESHOLD in
+ * memutils.h to agree.
  *--------------------
  */
 
@@ -385,7 +386,11 @@ AllocSetContextCreate(MemoryContext parent,
 	 * allocChunkLimit a power of two, because the requested and
 	 * actually-allocated sizes of any chunk must be on the same side of the
 	 * limit, else we get confused about whether the chunk is "big".
+	 *
+	 * Also, allocChunkLimit must not exceed ALLOCSET_SEPARATE_THRESHOLD.
 	 */
+	Assert(ALLOC_CHUNK_LIMIT == ALLOCSET_SEPARATE_THRESHOLD);
+
 	context->allocChunkLimit = ALLOC_CHUNK_LIMIT;
 	while (context->allocChunkLimit >
 		   (Size) (maxBlockSize - ALLOC_BLOCKHDRSZ - ALLOC_CHUNKHDRSZ))
diff --git a/src/backend/utils/sort/tuplesort.c b/src/backend/utils/sort/tuplesort.c
index 4900796e405..24591e3dc8a 100644
--- a/src/backend/utils/sort/tuplesort.c
+++ b/src/backend/utils/sort/tuplesort.c
@@ -531,7 +531,14 @@ tuplesort_begin_common(int workMem, bool randomAccess)
 	state->tapeset = NULL;
 
 	state->memtupcount = 0;
-	state->memtupsize = 1024;	/* initial guess */
+
+	/*
+	 * Initial size of array must be more than ALLOCSET_SEPARATE_THRESHOLD;
+	 * see comments in grow_memtuples().
+	 */
+	state->memtupsize = Max(1024,
+						ALLOCSET_SEPARATE_THRESHOLD / sizeof(SortTuple) + 1);
+
 	state->memtuples = (SortTuple *) palloc(state->memtupsize * sizeof(SortTuple));
 
 	USEMEM(state, GetMemoryChunkSpace(state->memtuples));
@@ -897,7 +904,7 @@ grow_memtuples(Tuplesortstate *state)
 				 state->memtupsize * sizeof(SortTuple));
 	USEMEM(state, GetMemoryChunkSpace(state->memtuples));
 	if (LACKMEM(state))
-		elog(ERROR, "unexpected out-of-memory situation during sort");
+		elog(ERROR, "unexpected out-of-memory situation in tuplesort");
 	return true;
 }
 
diff --git a/src/backend/utils/sort/tuplestore.c b/src/backend/utils/sort/tuplestore.c
index 55c8eca7957..4dfc5f6c5f3 100644
--- a/src/backend/utils/sort/tuplestore.c
+++ b/src/backend/utils/sort/tuplestore.c
@@ -261,7 +261,14 @@ tuplestore_begin_common(int eflags, bool interXact, int maxKBytes)
 
 	state->memtupdeleted = 0;
 	state->memtupcount = 0;
-	state->memtupsize = 1024;	/* initial guess */
+
+	/*
+	 * Initial size of array must be more than ALLOCSET_SEPARATE_THRESHOLD;
+	 * see comments in grow_memtuples().
+	 */
+	state->memtupsize = Max(16384 / sizeof(void *),
+							ALLOCSET_SEPARATE_THRESHOLD / sizeof(void *) + 1);
+
 	state->memtuples = (void **) palloc(state->memtupsize * sizeof(void *));
 
 	USEMEM(state, GetMemoryChunkSpace(state->memtuples));
@@ -644,6 +651,8 @@ tuplestore_puttuple_common(Tuplestorestate *state, void *tuple)
 						repalloc(state->memtuples,
 								 state->memtupsize * sizeof(void *));
 					USEMEM(state, GetMemoryChunkSpace(state->memtuples));
+					if (LACKMEM(state))
+						elog(ERROR, "unexpected out-of-memory situation in tuplestore");
 				}
 			}
 
diff --git a/src/include/utils/memutils.h b/src/include/utils/memutils.h
index d4a9d46dae7..c76d1aad2f4 100644
--- a/src/include/utils/memutils.h
+++ b/src/include/utils/memutils.h
@@ -136,4 +136,12 @@ extern MemoryContext AllocSetContextCreate(MemoryContext parent,
 #define ALLOCSET_SMALL_INITSIZE  (1 * 1024)
 #define ALLOCSET_SMALL_MAXSIZE	 (8 * 1024)
 
+/*
+ * Threshold above which a request in an AllocSet context is certain to be
+ * allocated separately (and thereby have constant allocation overhead).
+ * Few callers should be interested in this, but tuplesort/tuplestore need
+ * to know it.
+ */
+#define ALLOCSET_SEPARATE_THRESHOLD  8192
+
 #endif   /* MEMUTILS_H */