This example uses OpenLayers v10.9.0. The latest is v.
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#### CartoDB source example

simple 7 openstreetmap 22 attribution 1

#####

Show european countries larger than

Example of a cartodb map.

A simple example with an anonymous cartodb map

##### main.js

    import Map from 'ol/Map.js';
    import View from 'ol/View.js';
    import TileLayer from 'ol/layer/Tile.js';
    import CartoDB from 'ol/source/CartoDB.js';
    import OSM from 'ol/source/OSM.js';

    const mapConfig = {
      'layers': [
        {
          'type': 'cartodb',
          'options': {
            'cartocss_version': '2.1.1',
            'cartocss': '#layer { polygon-fill: #F00; }',
          },
        },
      ],
    };

    function setArea(n) {
      mapConfig.layers[0].options.sql =
        'select * from european_countries_e where area > ' + n;
    }
    const areaSelect = document.getElementById('country-area');
    setArea(areaSelect.value);

    const cartoDBSource = new CartoDB({
      account: 'documentation',
      config: mapConfig,
    });

    areaSelect.addEventListener('change', function () {
      setArea(this.value);
      cartoDBSource.setConfig(mapConfig);
    });

    const map = new Map({
      layers: [
        new TileLayer({
          source: new OSM(),
        }),
        new TileLayer({
          source: cartoDBSource,
        }),
      ],
      target: 'map',
      view: new View({
        center: [8500000, 8500000],
        zoom: 2,
      }),
    });

##### index.html

    <!DOCTYPE html>
    <html lang="en">
      <head>
        <meta charset="UTF-8">
        <title>CartoDB source example</title>
        <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/bootstrap@5.2.0/dist/css/bootstrap.min.css">
        <link rel="stylesheet" href="node_modules/ol/ol.css">
        <style>
          .map {
            width: 100%;
            height: 400px;
          }
        </style>
      </head>
      <body>
        <div id="map" class="map"></div>
        <form class="row">
          <div class="col-auto">
            <div class="input-group">
              <label for="country-area" class="input-group-text">Show european countries larger than</label>
              <select id="country-area" class="form-select">
                <option value="0" default>0 ㎢</option>
                <option value="5000">5000 ㎢</option>
                <option value="10000">10000 ㎢</option>
                <option value="50000">50000 ㎢</option>
                <option value="100000">100000 ㎢</option>
              </select>
            </div>
          </div>
        </form>

        <script type="module" src="main.js"></script>
      </body>
    </html>

##### package.json

    {
      "name": "cartodb",
      "dependencies": {
        "ol": "10.9.0"
      },
      "devDependencies": {
        "vite": "^3.2.3"
      },
      "scripts": {
        "start": "vite",
        "build": "vite build"
      }
    }