Starting with 2, integers are added, in order, as vertices to a graph. Any two vertices are joined if one of them is a factor of the other. What is the maximum integer if none of the edges are allowed to cross? https://twitter.com/mathstechnology/status/1407585626015338499

integers_factor_graph.js

//numbers 2-23
var run = async _ => {
  let canvas = $("<canvas width=1000 height=800></canvas>")[0]
  $('body').empty().append(canvas)

  const ctx = canvas.getContext('2d');

  const get_random = (a, b) => {
    return (b - a) * Math.random() + a;
  }

  const draw_circle = (x, y, r) => {
    // Draw the ellipse
    ctx.beginPath();
    ctx.ellipse(x, y, r, r, Math.PI / 4, 0, 2 * Math.PI); //
    ctx.stroke();
  }

  const draw_text = (x, y, n) => {

    ctx.font = 'bold 24px monospace';
    ctx.fillStyle = 'red';
    ctx.fillText(`${n}`, x - 5, y + 5);
  }

  const draw_line = (p1, p2) => {
    // Draw the ellipse
    ctx.beginPath();
    ctx.moveTo(p1[0], p1[1])
    ctx.lineTo(p2[0], p2[1])
    ctx.stroke();
  }

  const check_cross = (line1, line2) => {
    let [x11, y11] = line1[0], [x12, y12] = line1[1],
      [x21, y21] = line2[0], [x22, y22] = line2[1];

    let xa = [x12 - x11, y12 - y11],
      xb = [x22 - x21, y22 - y21];

    if (Math.abs(xa[0] * xb[1] - xa[1] * xb[0]) < 1e-6) return false;

    let x1min = Math.min(x11, x12),
      x1max = Math.max(x11, x12),
      x2min = Math.min(x21, x22),
      x2max = Math.max(x21, x22),

      k1 = xa[1] / xa[0],
      b1 = y11 - x11 * k1,
      k2 = xb[1] / xb[0],
      b2 = y21 - x21 * k2;

    let cross_x = (b2 - b1) / (k1 - k2);
    //console.log(cross_x)

    if (cross_x > x1min + 0.1 && cross_x < x1max - 0.1 &&
      cross_x > x2min + 0.1 && cross_x < x2max - 0.1) return true;

    return false;
  }

  console.log(check_cross([
    [0, 0],
    [100, 100]
  ], [
    [0, 2],
    [100, 102]
  ])); //false
  console.log(check_cross([
    [0, 0],
    [100, 100]
  ], [
    [2, 2],
    [102, 102]
  ])); //false
  console.log(check_cross([
    [0, 0],
    [100, 100]
  ], [
    [0, 100],
    [100, 0]
  ])); //true

  console.log(check_cross([
    [0, 0],
    [100, 100]
  ], [
    [0, 100],
    [50, 55]
  ])); //false

  const test_once = (N, show_circle = false) => {
    let lines = [],
      points = new Map();

    if (show_circle) ctx.clearRect(0, 0, 1000, 800);

    let findall = true;
    for (let n = 2; n <= N; n++) {

      let find = false;
      //let lines_back = JSON.parse(JSON.stringify(lines))
      const lines_length = lines.length;
      for (let i = 0; i < n * n; i++) { //every point try n*n times
        //lines = JSON.parse(JSON.stringify(lines_back));
        lines.length = lines_length;

        let x = get_random(100, 900),
          y = get_random(100, 700);

        if (show_circle) {
          draw_circle(x, y, 30)
          draw_text(x, y, n)
        }

        points.set(n, [x, y])

        let cross = false;
        for (let k = 2; k < n; k++) { //check cross of divide

          if (n % k === 0) {
            let line1 = [points.get(k), points.get(n)];

            for (let line2 of lines) {
              if (check_cross(line1, line2)) {
                cross = true;
                break;
              };
            }
            if (cross) break;


            if (show_circle) draw_line(line1[0], line1[1])

            lines.push(line1)
          }

        }
        if (cross) continue;

        find = true;
        break;
      }

      if (!find) {
        findall = false;
        break;
      }

    }

    if (!findall) return false;

    ctx.clearRect(0, 0, 1000, 800);
    for (let n = 2; n <= N; n++) {
      let [x, y] = points.get(n);

      draw_circle(x, y, 30)
      draw_text(x, y, n)
    }
    for (let line of lines) {
      draw_line(line[0], line[1])
    }

    return true;
  }

  function sleep(ms) {
    return new Promise(resolve => setTimeout(resolve, ms));
  }

  let show_circle = false;
  for (let i = 0; i < 1e6; i++) {
    if (i % 1E2 === 0) console.log(`find ${i} times`);

    if (i % 1E2 === 0) {
      //show_circle = true;
      //await sleep(100)
    } else show_circle = false;

    if (test_once(23, show_circle)) {
      break;
    }
  }
}

run()

integers_factor_graph2.js

//any numbers
var run = async _ => {
  let canvas = $("<canvas width=1000 height=800></canvas>")[0]
  $('body').empty().append(canvas)

  const ctx = canvas.getContext('2d');

  const get_random = (a, b) => {
    return (b - a) * Math.random() + a;
  }

  const draw_circle = (x, y, r) => {
    // Draw the ellipse
    ctx.beginPath();
    ctx.ellipse(x, y, r, r, Math.PI / 4, 0, 2 * Math.PI); //
    ctx.stroke();
  }

  const draw_text = (x, y, n) => {

    ctx.font = 'bold 24px monospace';
    ctx.fillStyle = 'red';
    ctx.fillText(`${n}`, x - 5, y + 5);
  }

  const draw_line = (p1, p2) => {
    // Draw the ellipse
    ctx.beginPath();
    ctx.moveTo(p1[0], p1[1])
    ctx.lineTo(p2[0], p2[1])
    ctx.stroke();
  }

  const check_cross = (line1, line2) => {
    let [x11, y11] = line1[0], [x12, y12] = line1[1],
      [x21, y21] = line2[0], [x22, y22] = line2[1];

    let xa = [x12 - x11, y12 - y11],
      xb = [x22 - x21, y22 - y21];

    //if (Math.abs(xa[0]) < 1e-6 || Math.abs(xb[0]) < 1e-6) return true;
    if (Math.abs(xa[0] * xb[1] - xa[1] * xb[0]) < 1e-6) return false;

    let x1min = Math.min(x11, x12),
      x1max = Math.max(x11, x12),
      x2min = Math.min(x21, x22),
      x2max = Math.max(x21, x22),

      k1 = xa[1] / xa[0],
      b1 = y11 - x11 * k1,
      k2 = xb[1] / xb[0],
      b2 = y21 - x21 * k2;

    let cross_x = (b2 - b1) / (k1 - k2);
    //console.log(cross_x)

    if (cross_x > x1min && cross_x < x1max &&
      cross_x > x2min && cross_x < x2max) return true;

    return false;
  }

  console.log(check_cross([
    [0, 0],
    [100, 100]
  ], [
    [0, 2],
    [100, 102]
  ])); //false
  console.log(check_cross([
    [0, 0],
    [100, 100]
  ], [
    [2, 2],
    [102, 102]
  ])); //false
  console.log(check_cross([
    [0, 0],
    [100, 100]
  ], [
    [0, 100],
    [100, 0]
  ])); //true

  console.log(check_cross([
    [0, 0],
    [100, 100]
  ], [
    [0, 100],
    [50, 55]
  ])); //false

  const test_once = (numbers, show_circle = false) => {
    let lines = [],
      points = new Map();

    if (show_circle) ctx.clearRect(0, 0, 1000, 800);

    let findall = true;
    for (let n of numbers) {

      let find = false;
      //let lines_back = JSON.parse(JSON.stringify(lines))
      const lines_length = lines.length;
      for (let i = 0; i < n * n; i++) { //every point try n*n times
        //lines = JSON.parse(JSON.stringify(lines_back));
        lines.length = lines_length;

        let x = get_random(100, 900),
          y = get_random(100, 700);

        if (show_circle) {
          draw_circle(x, y, 30)
          draw_text(x, y, n)
        }

        points.set(n, [x, y])

        let cross = false;
        for (let k of numbers) { //check cross of divide
          if (k >= n) break;

          if (n % k === 0) {
            let line1 = [points.get(k), points.get(n)];

            for (let line2 of lines) {
              if (check_cross(line1, line2)) {
                cross = true;
                break;
              };
            }
            if (cross) break;


            if (show_circle) draw_line(line1[0], line1[1])

            lines.push(line1)
          }

        }
        if (cross) continue;

        find = true;
        break;
      }

      if (!find) {
        findall = false;
        break;
      }

    }

    if (!findall) return false;

    ctx.clearRect(0, 0, 1000, 800);
    for (let n of numbers) {
      let [x, y] = points.get(n);

      draw_circle(x, y, 20)
      draw_text(x, y, n)
    }
    for (let line of lines) {
      draw_line(line[0], line[1])
    }

    return true;
  }

  function sleep(ms) {
    return new Promise(resolve => setTimeout(resolve, ms));
  }

  let show_circle = false;

  const numbers = [];
  for (let i = 20; i <= 100; i++) {
    numbers.push(i);
  }

  for (let i = 0; i < 1e6; i++) {
    if (i % 1E2 === 0) console.log(`find ${i} times`);

    if (i % 1E2 === 0) {
      //show_circle = true;
      //await sleep(100)
    } else show_circle = false;

    if (test_once(numbers, show_circle)) {
      break;
    }
  }
}

run()