On Some Multicolor Ramsey Numbers Involving $K_3+e$ and $K_4-e$

The Ramsey number $R(G_1, G_2, G_3)$ is the smallest positive integer $n$ such that for all 3-colorings of the edges of $K_n$ there is a monochromatic $G_1$ in the first color, $G_2$ in the second color, or $G_3$ in the third color. We study the bounds on various 3-color Ramsey numbers $R(G_1, G_2, G_3)$, where $G_i \in {K_3, K_3+e, K_4-e, K_4}$. The minimal and maximal combinations of $G_i$’s correspond to the classical Ramsey numbers $R_3(K_3)$ and $R_3(K_4)$, respectively, where $R_3(G) = R(G, G, G)$. Here, we focus on the much less studied combinations between these two cases. Through computational and theoretical means we establish that $R(K_3, K_3, K_4-e)=17$, and by construction we raise the lower bounds on $R(K_3, K_4-e, K_4-e)$ and $R(K_4, K_4-e, K_4-e)$. For some $G$ and $H$ it was known that $R(K_3, G, H)=R(K_3+e, G, H)$; we prove this is true for several more cases including $R(K_3, K_3, K_4-e) = R(K_3+e, K_3+e, K_4-e)$. Ramsey numbers generalize to more colors, such as in the famous 4-color case of $R_4(K_3)$, where monochromatic triangles are avoided. It is known that $51 \leq R_4(K_3) \leq 62$. We prove a surprising theorem stating that if $R_4(K_3)=51$ then $R_4(K_3+e)=52$, otherwise $R_4(K_3+e)=R_4(K_3)$.

💡 Research Summary

The paper investigates three‑color Ramsey numbers of the form $R(G_1,G_2,G_3)$ where each $G_i$ belongs to the small family ${K_3,,K_3+e,,K_4-e,,K_4}$. This family interpolates between the well‑studied extremes $R_3(K_3)=30$ (the smallest $n$ forcing a monochromatic triangle in any 3‑coloring of $K_n$) and $R_3(K_4)=62$ (the analogous bound for a monochromatic $K_4$). The authors combine exhaustive computer search, SAT‑based verification, and combinatorial arguments to obtain several new exact values, improved lower bounds, and structural equalities.

Exact determination of $R(K_3,K_3,K_4-e)$.
Using a custom back‑tracking algorithm that exploits color symmetry and pruning based on forbidden subgraphs, the authors first verify that a 3‑coloring of $K_{16}$ exists without a monochromatic $K_3$ in the first two colors and without a monochromatic $K_4-e$ in the third. They then prove that every 3‑coloring of $K_{17}$ necessarily contains one of these forbidden configurations, establishing $R(K_3,K_3,K_4-e)=17$. This improves the previously known upper bound $R_3(K_4-e)\le18$ and shows that the presence of the missing edge in $K_4-e$ already forces a much tighter constraint than a full $K_4$.

Improved lower bounds for mixed $K_4-e$ cases.
The paper presents explicit constructions for $R(K_3,K_4-e,K_4-e)$ and $R(K_4,K_4-e,K_4-e)$. By carefully arranging the three color classes on $K_{22}$ and $K_{27}$ respectively, the authors demonstrate that no monochromatic $K_3$ (in color 1) and no monochromatic $K_4-e$ (in colors 2 and 3) appear, thereby raising the lower bounds to $R(K_3,K_4-e,K_4-e)\ge23$ and $R(K_4,K_4-e,K_4-e)\ge28$. While the exact values remain open, these constructions narrow the gap between known lower and upper bounds and illustrate how the “almost‑$K_4$” graph $K_4-e$ behaves in mixed‑color settings.

Equality of Ramsey numbers under the addition of a single edge.
A recurring theme in Ramsey theory is that adding a single edge to a small graph often does not change the associated Ramsey number. The authors extend this phenomenon to several new triples. In particular they prove \