feat: add invitations badge with available invites

frontend/lib/screens/home_screen.dart

@@ -2,6 +2,7 @@ import 'package:flutter/material.dart';
 import 'pages/feed_page.dart';
 import 'pages/invitations_page.dart';
 import 'pages/profile_page.dart';
+import '../services/invitations_service.dart';
 
 class HomeScreen extends StatefulWidget {
   @override
@@ -10,9 +11,70 @@ class HomeScreen extends StatefulWidget {
 
 class _HomeScreenState extends State<HomeScreen> {
   int _currentIndex = 0;
+  int _availableInvitationsCount = 0;
 
   final List<Widget> _pages = [FeedPage(), InvitationsPage(), ProfilePage()];
 
+  @override
+  void initState() {
+    super.initState();
+    // Start polling and listen to invitations stream
+    InvitationsService.startPolling();
+    _listenToInvitations();
+  }
+
+  @override
+  void dispose() {
+    InvitationsService.stopPolling();
+    super.dispose();
+  }
+
+  void _listenToInvitations() {
+    InvitationsService.getInvitationsStream().listen((invitationsData) {
+      if (mounted) {
+        setState(() {
+          _availableInvitationsCount = invitationsData.available.length;
+        });
+      }
+    });
+  }
+
+  Widget _buildInvitationsBadge() {
+    if (_availableInvitationsCount == 0) {
+      return Icon(Icons.mail);
+    }
+    
+    return Stack(
+      children: [
+        Icon(Icons.mail),
+        Positioned(
+          right: 0,
+          top: 0,
+          child: Container(
+            padding: EdgeInsets.all(1),
+            decoration: BoxDecoration(
+              color: Colors.red,
+              borderRadius: BorderRadius.circular(10),
+            ),
+            constraints: BoxConstraints(
+              minWidth: 16,
+              minHeight: 16,
+            ),
+            child: Text(
+              _availableInvitationsCount > 99 ? '99+' : '$_availableInvitationsCount',
+              style: TextStyle(
+                color: Colors.white,
+                fontSize: 10,
+                fontWeight: FontWeight.bold,
+              ),
+              textAlign: TextAlign.center,
+            ),
+          ),
+        ),
+      ],
+    );
+  }
+
   @override
   Widget build(BuildContext context) {
     return PopScope(
@@ -32,7 +94,7 @@ class _HomeScreenState extends State<HomeScreen> {
           items: [
             BottomNavigationBarItem(icon: Icon(Icons.home), label: 'Feed'),
             BottomNavigationBarItem(
-              icon: Icon(Icons.mail),
+              icon: _buildInvitationsBadge(),
               label: 'Invitations',
             ),
             BottomNavigationBarItem(icon: Icon(Icons.person), label: 'Profile'),