known

  • paths $P_n$
  • $ORN \leq SRN$
  • for an infinite number of cliques the $ORN(K_n) \leq SRN(K_n)$
  • Haxell lowerbound on SRN of trees is $|T_1|\Delta(T_1)+|T_2|\Delta(T_2)$

thesis

  • infinite family of trees for which ORN is asymptotically smaller than SRN
  • cycles $C_n$ in $O(n)$
  • spiders $S_{k,l}$ in $O(k^2 l)$
  • exact value for $\tilde{r}(C_3,S_n)$ on connected graphs is $3n-1$

generally

  • induced paths $P_n$ in $O(n)$
    • implies induced cycles, spiders (and more?)

nsfocs

  • paths on 3 colors in $O(n^2)$

kamak

  • backtracked $\tilde{r}(C_3,C_3,C_3)$ to be less than 10
  • two stars $S_n$ and $S_m$ connected with one edge in $O(n+m)$
  • lowerbound for trees is $\Omega(\Delta(T) VC(T))$
  • lowerbound on spiders $S_{k,l}$ is ~ $\frac{k^2 l}{4}$
  • path and triangle $\tilde{r}(P_n,C_3)$ in $O(n)$
  • path and triangles on three colors $\tilde{r}(P_n,C_3,C_3)$ in $O(n)$
  • path $P_n$ alterating vertices of two sets in $O(n)$
  • cycle with one path connected to it
  • centipides $P_{n,k}$ in $O(k^2 n)$
  • paths on $k$ colors in $O(n^{k-1})$
  • path of length $N$ or $N+1$ with $C_3$ connected to each end vertex

po kamaku

  • path with a graph H on each vertex cca $O(n \widetilde{r}(H^2))$
  • cycle with a graph H on each vertex or a centipide
  • broken centipide (with stars on a least n/2 stars)