diff --git a/compiler/src/dotty/tools/dotc/core/Denotations.scala b/compiler/src/dotty/tools/dotc/core/Denotations.scala
index 15eb8c1e65a9..e2568ad0461d 100644
--- a/compiler/src/dotty/tools/dotc/core/Denotations.scala
+++ b/compiler/src/dotty/tools/dotc/core/Denotations.scala
@@ -494,8 +494,6 @@ object Denotations {
          *      boundary of sym1. For protected access, we count the enclosing
          *      package as access boundary.
          *   5. sym1 is a method but sym2 is not.
-         *   6. sym1 is a non-polymorphic method but sym2 is a polymorphic method.
-         *      (to be consistent with infoMeet, see #4819)
          *  The aim of these criteria is to give some disambiguation on access which
          *   - does not depend on textual order or other arbitrary choices
          *   - minimizes raising of doubleDef errors
@@ -510,7 +508,6 @@ object Denotations {
                 accessBoundary(sym2).isProperlyContainedIn(accessBoundary(sym1)) ||
                 sym2.is(Bridge) && !sym1.is(Bridge) ||
                 sym1.is(Method) && !sym2.is(Method)) ||
-                sym1.info.isInstanceOf[MethodType] && sym2.info.isInstanceOf[PolyType] ||
               sym1.info.isErroneous)
 
         /** Sym preference provided types also override */
@@ -1071,8 +1068,12 @@ object Denotations {
 
     final def matches(other: SingleDenotation)(implicit ctx: Context): Boolean = {
       val d = signature.matchDegree(other.signature)
-      d == Signature.FullMatch ||
-      d >= Signature.ParamMatch && info.matches(other.info)
+      (// fast path: signatures are the same and neither denotation is a PolyType
+       // For polytypes, signatures alone do not tell us enough to be sure about matching.
+       d == Signature.FullMatch &&
+       !infoOrCompleter.isInstanceOf[PolyType] && !other.infoOrCompleter.isInstanceOf[PolyType]
+       ||
+       d >= Signature.ParamMatch && info.matches(other.info))
     }
 
     def mapInherited(ownDenots: PreDenotation, prevDenots: PreDenotation, pre: Type)(implicit ctx: Context): SingleDenotation =
diff --git a/compiler/src/dotty/tools/dotc/core/Types.scala b/compiler/src/dotty/tools/dotc/core/Types.scala
index 710edf35d78e..00ebbf7b4deb 100644
--- a/compiler/src/dotty/tools/dotc/core/Types.scala
+++ b/compiler/src/dotty/tools/dotc/core/Types.scala
@@ -1916,8 +1916,6 @@ object Types {
       setDenot(memberDenot(name, allowPrivate = !symbol.exists || symbol.is(Private)))
 
     private def setDenot(denot: Denotation)(implicit ctx: Context): Unit = {
-      if (ctx.isAfterTyper)
-        assert(!denot.isOverloaded || ctx.mode.is(Mode.Printing), this)
       if (Config.checkNoDoubleBindings)
         if (ctx.settings.YnoDoubleBindings.value)
           checkSymAssign(denot.symbol)
diff --git a/compiler/src/dotty/tools/dotc/typer/TypeAssigner.scala b/compiler/src/dotty/tools/dotc/typer/TypeAssigner.scala
index 17fc86a6c571..317d3f762696 100644
--- a/compiler/src/dotty/tools/dotc/typer/TypeAssigner.scala
+++ b/compiler/src/dotty/tools/dotc/typer/TypeAssigner.scala
@@ -381,65 +381,74 @@ trait TypeAssigner {
   }
 
   def assignType(tree: untpd.TypeApply, fn: Tree, args: List[Tree])(implicit ctx: Context): TypeApply = {
-    val ownType = fn.tpe.widen match {
+    def fail = tree.withType(errorType(err.takesNoParamsStr(fn, "type "), tree.pos))
+    fn.tpe.widen match {
       case pt: TypeLambda =>
-        val paramNames = pt.paramNames
-        if (hasNamedArg(args)) {
-          val paramBoundsByName = paramNames.zip(pt.paramInfos).toMap
-
-          // Type arguments which are specified by name (immutable after this first loop)
-          val namedArgMap = new mutable.HashMap[Name, Type]
-          for (NamedArg(name, arg) <- args)
-            if (namedArgMap.contains(name))
-              ctx.error(DuplicateNamedTypeParameter(name), arg.pos)
-            else if (!paramNames.contains(name))
-              ctx.error(UndefinedNamedTypeParameter(name, paramNames), arg.pos)
-            else
-              namedArgMap(name) = arg.tpe
-
-          // Holds indexes of non-named typed arguments in paramNames
-          val gapBuf = new mutable.ListBuffer[Int]
-          def nextPoly(idx: Int) = {
-            val newIndex = gapBuf.length
-            gapBuf += idx
-            // Re-index unassigned type arguments that remain after transformation
-            pt.paramRefs(newIndex)
-          }
+        tree.withType {
+          val paramNames = pt.paramNames
+          if (hasNamedArg(args)) {
+            val paramBoundsByName = paramNames.zip(pt.paramInfos).toMap
+
+            // Type arguments which are specified by name (immutable after this first loop)
+            val namedArgMap = new mutable.HashMap[Name, Type]
+            for (NamedArg(name, arg) <- args)
+              if (namedArgMap.contains(name))
+                ctx.error(DuplicateNamedTypeParameter(name), arg.pos)
+              else if (!paramNames.contains(name))
+                ctx.error(UndefinedNamedTypeParameter(name, paramNames), arg.pos)
+              else
+                namedArgMap(name) = arg.tpe
+
+            // Holds indexes of non-named typed arguments in paramNames
+            val gapBuf = new mutable.ListBuffer[Int]
+            def nextPoly(idx: Int) = {
+              val newIndex = gapBuf.length
+              gapBuf += idx
+              // Re-index unassigned type arguments that remain after transformation
+              pt.paramRefs(newIndex)
+            }
 
-          // Type parameters after naming assignment, conserving paramNames order
-          val normArgs: List[Type] = paramNames.zipWithIndex.map { case (pname, idx) =>
-            namedArgMap.getOrElse(pname, nextPoly(idx))
-          }
+            // Type parameters after naming assignment, conserving paramNames order
+            val normArgs: List[Type] = paramNames.zipWithIndex.map { case (pname, idx) =>
+              namedArgMap.getOrElse(pname, nextPoly(idx))
+            }
 
-          val transform = new TypeMap {
-            def apply(t: Type) = t match {
-              case TypeParamRef(`pt`, idx) => normArgs(idx)
-              case _ => mapOver(t)
+            val transform = new TypeMap {
+              def apply(t: Type) = t match {
+                case TypeParamRef(`pt`, idx) => normArgs(idx)
+                case _ => mapOver(t)
+              }
+            }
+            val resultType1 = transform(pt.resultType)
+            if (gapBuf.isEmpty) resultType1
+            else {
+              val gaps = gapBuf.toList
+              pt.derivedLambdaType(
+                gaps.map(paramNames),
+                gaps.map(idx => transform(pt.paramInfos(idx)).bounds),
+                resultType1)
             }
           }
-          val resultType1 = transform(pt.resultType)
-          if (gapBuf.isEmpty) resultType1
           else {
-            val gaps = gapBuf.toList
-            pt.derivedLambdaType(
-              gaps.map(paramNames),
-              gaps.map(idx => transform(pt.paramInfos(idx)).bounds),
-              resultType1)
+            val argTypes = args.tpes
+            if (sameLength(argTypes, paramNames)) pt.instantiate(argTypes)
+            else wrongNumberOfTypeArgs(fn.tpe, pt.typeParams, args, tree.pos)
           }
         }
-        else {
-          val argTypes = args.tpes
-          if (sameLength(argTypes, paramNames)) pt.instantiate(argTypes)
-          else wrongNumberOfTypeArgs(fn.tpe, pt.typeParams, args, tree.pos)
-        }
       case err: ErrorType =>
-        err
+        tree.withType(err)
+      case ref: TermRef if ref.isOverloaded =>
+        val disambiguated = ref.denot.suchThat(_.info.isInstanceOf[PolyType])
+        if (disambiguated.exists) {
+          val fn1 = fn.withType(ref.withDenot(disambiguated))
+          val tree1 = untpd.cpy.TypeApply(tree)(fn1, args)
+          assignType(tree1, fn1, args)
+        }
+        else fail
       case _ =>
         //println(i"bad type: $fn: ${fn.symbol} / ${fn.symbol.isType} / ${fn.symbol.info}") // DEBUG
-        errorType(err.takesNoParamsStr(fn, "type "), tree.pos)
+        fail
     }
-
-    tree.withType(ownType)
   }
 
   def assignType(tree: untpd.Typed, tpt: Tree)(implicit ctx: Context): Typed =
diff --git a/tests/neg/i4819.scala b/tests/neg/i4819.scala
index 224682fc000c..8a9f89d45a13 100644
--- a/tests/neg/i4819.scala
+++ b/tests/neg/i4819.scala
@@ -7,10 +7,6 @@ trait Two[Y <: Foo] {
 }
 
 class Foo extends One[Foo] with Two[Foo] {
-  concat(0) // OK
-
-  // TODO: This does not typecheck because the polymorphic overload is masked
-  // (we merge the denotations for both overloads into one and always prefer
-  // MethodType to PolyType, instead we should return a MultiDenotation). See #4819.
-  concat[Int](0) // error (that should actually not be an error)
+  concat(0) // error: ambiguous overload
+  concat[Int](0) // OK
 }
diff --git a/tests/neg/i5445.scala b/tests/neg/i5445.scala
new file mode 100644
index 000000000000..2cbe22d4ddd2
--- /dev/null
+++ b/tests/neg/i5445.scala
@@ -0,0 +1,6 @@
+object Test {
+
+  trait A { def polymorphic[x]: Int }
+  val a = new A { val polymorphic = Unit } // error: object creation impossible
+
+}
\ No newline at end of file
diff --git a/tests/neg/i5578.scala b/tests/neg/i5578.scala
new file mode 100644
index 000000000000..b6a1ec713aef
--- /dev/null
+++ b/tests/neg/i5578.scala
@@ -0,0 +1,12 @@
+trait P[A]{
+  def a[T]: A
+}
+class C extends P[Int]{ // error: class C needs to be abstract
+  def a = 1
+}
+object O{
+  def main(args: Array[String]) = {
+    val p: P[Int] = new C
+    println(p.a)
+  }
+}
\ No newline at end of file
diff --git a/tests/pos/i4819.scala b/tests/pos/i4819.scala
index 508c579a8753..0494e9adfa55 100644
--- a/tests/pos/i4819.scala
+++ b/tests/pos/i4819.scala
@@ -7,6 +7,12 @@ trait Two[Y <: Foo] {
 }
 
 class Foo extends One[Foo] with Two[Foo] {
-  concat(0) // OK
+  concat[Int](0) // OK
   // See also tests/neg/i4819.scala
 }
+
+class Bar extends One[String] with Two[Foo] {
+  val x: String = concat(0)
+  val y = concat[Int](0)
+  val z: Foo = concat(0)
+}
\ No newline at end of file