Mathematics · Ramsey theory Small Ramsey Numbers
Whether any $(5,5,n)$-graph exists for $n \in \{43,44,45\}$ ($43 \leq R(5,5) \leq 46$) or outside the $656$ known $(5,5,42)$-graphs with $|\operatorname{Aut}(G)
A Goodman count splits the existence question for a $(5,5,45)$ graph, whose nonexistence would give $R(5,5)\le 45$, into two cases. Either some pair of vertices
An R-graph is a 10-regular graph on 22 vertices with no $K_4$ and no independent set of size 5. If $G$ is a $(5,5)$-good $\mathrm{srg}(45,22,10,11)$ and $u$ is
Whether any $(5,5,42)$-graph exists beyond the $656$ known graphs ($328$ complementary pairs) and whether any $(5,5,43)$-graph exists ($43 \leq R(5,5) \leq 46$)
Can small common neighborhoods constrain how saturated monochromatic triangles coexist in a hypothetical $(5,5,45)$ graph? Call a monochromatic triangle saturat
A graph is (5,5)-good if it has neither a clique nor an independent set of order five. Building on Calder's accepted prime-automorphism exclusions and the class
Let $X$ be the local graph at a vertex of a strongly regular graph $\mathrm{srg}(45,22,10,11)$. A $(5,5)$-good such graph would show $R(5,5)=46$; excluding it i
Let G be a graph on 45 vertices with no clique or independent set of order five. Building on the published solvability theorem, we prove that if five divides th
Let $u$ be a vertex of a strongly regular graph $\mathrm{srg}(45,22,10,11)$, with local graph $X$ on the 22 neighbours of $u$. A $(5,5)$-good such graph, with n
We study faithful cyclic order-$6$ ($C_6 = \langle g \rangle \cong C_3 \times C_2$) automorphisms of two-colored $(K_5, I_5)$-free graphs across Orders $N \in \
We resolve the faithful cyclic order-$9$ ($C_9 = \langle g \rangle$) automorphism question for two-colored $(K_5, I_5)$-free graphs across all four open vertex
For a graph containing neither a clique nor an independent set of order five, we prove that every three-subgroup of its automorphism group has order at most nin
We study $C_3$-equivariant neighbourhood and two-block gluing for $(5,5)$-free graphs at orders $N \in \{42,43\}$, where a $(K_4,I_5)$-free source $A$ on $r_A$
We prove that every graph on $43$ vertices that is Seidel-switching-equivalent to a claw-free graph contains a clique or an independent set of order five. The r
