The cloning construction for R(3,3)

Clone one vertex. Color one final edge.

Begin with a triangle-free red–blue coloring of K5: a red 5-cycle and its blue complement. Clone vertex 1, leave e = 11′ uncolored, then choose its color.

Three steps in the cloning construction

01 · start below the threshold A target-free coloring of K5

The red cycle and blue complementary cycle contain no monochromatic triangle.

02 · clone vertex 1 Copy every incident color

Give 1′ the same colored neighbors as 1 and leave only e = 11′ missing.

03 · color the final edge Every new triangle passes through e

Red or blue, the last edge creates exactly two monochromatic triangles.

Follow the construction

Move from the original K5 to either completion of the cloned graph.

Construction state

Step 2 · clone vertex 1

K6 − e is still target-free.

The colored edges from 1′ copy the corresponding edges from 1; only e = 11′ is missing.

Triangle-free red-blue coloring of K6 minus one edge Vertex 1 prime is a clone of vertex 1. Their connecting edge is uncolored, and the graph has no monochromatic triangles.
red edge blue edge uncolored final edge cloned vertex

Copy-by-copy correspondence

The clone has the same colored neighbors as vertex 1.

Before e is colored, those matching neighborhoods do not close a triangle. Choose red or blue above to see the two witnesses.

red neighbors Nred(1) = {2, 5}

Both red incidences are copied from 1 to 1′.

blue neighbors Nblue(1) = {3, 4}

Both blue incidences are copied from 1 to 1′.