# Results in ORN

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)