.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/combinatorics/extremal/plot_01_ramsey_r33.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_combinatorics_extremal_plot_01_ramsey_r33.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_combinatorics_extremal_plot_01_ramsey_r33.py:


Ramsey's theorem: R(3,3) = 6
==================================

Searches every red/blue coloring of the edges of K_5 and K_6. K_5 has
colorings with no single-colored triangle, such as the pentagon and
pentagram, but every one of the 32768 colorings of K_6 contains one.

.. GENERATED FROM PYTHON SOURCE LINES 11-18

.. code-block:: Python

    from itertools import combinations

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.combinatorics import count_triangle_free_colorings, has_monochromatic_triangle








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Exhaustive search
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-26

.. code-block:: Python


    for n in range(3, 7):
        total = 2 ** (n * (n - 1) // 2)
        print(f"K_{n}: {count_triangle_free_colorings(n)} of {total} colorings avoid a monochromatic triangle")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    K_3: 6 of 8 colorings avoid a monochromatic triangle
    K_4: 18 of 64 colorings avoid a monochromatic triangle
    K_5: 12 of 1024 colorings avoid a monochromatic triangle
    K_6: 0 of 32768 colorings avoid a monochromatic triangle




.. GENERATED FROM PYTHON SOURCE LINES 27-29

The extremal coloring of K_5
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-40

.. code-block:: Python


    coloring = {(i, j): int((j - i) % 5 in (1, 4)) for i, j in combinations(range(5), 2)}
    print(f"pentagon/pentagram coloring has a monochromatic triangle: {has_monochromatic_triangle(5, coloring)}")
    points = np.array([(np.sin(2 * np.pi * k / 5), np.cos(2 * np.pi * k / 5)) for k in range(5)])
    fig, ax = plt.subplots()
    for (i, j), color in coloring.items():
        ax.plot(*points[[i, j]].T, color=("tab:red" if color else "tab:blue"), lw=2)
    ax.plot(*points.T, "ko")
    ax.set_aspect("equal")
    ax.axis("off")
    ax.set_title("K_5 with no one-colored triangle")



.. image-sg:: /api/gallery/combinatorics/extremal/images/sphx_glr_plot_01_ramsey_r33_001.png
   :alt: K_5 with no one-colored triangle
   :srcset: /api/gallery/combinatorics/extremal/images/sphx_glr_plot_01_ramsey_r33_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    pentagon/pentagram coloring has a monochromatic triangle: False

    Text(0.5, 1.0, 'K_5 with no one-colored triangle')




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.048 seconds)


.. _sphx_glr_download_api_gallery_combinatorics_extremal_plot_01_ramsey_r33.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_ramsey_r33.ipynb <plot_01_ramsey_r33.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_ramsey_r33.py <plot_01_ramsey_r33.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_ramsey_r33.zip <plot_01_ramsey_r33.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_