Chi_c of subgraphs

Given a rational number p/q, we say a rational number p'/q' is unavoidable by p/q if every graph G with chi_c(G) = p/q contains a subgraph H with chi_c(H) = p'/q'.

Problem: Determine the set of rational numbers unavoidable by p/q.

Suppose (p, q)=1, i.e., p and q are coprime. Let p', q' be the unique integers such that 0 < p' < p and pq'-p'q=1. We call p'/q' the lower parent of p/q, and denote it by p_l(p/q).

Question 1: Is true that for every rational p/q,  p_l(p/q) is unavoidable by p/q ?

In particular, the following simplest special case of the above question remain open.

Question 2:  Is it true that 5/2 is unavoidable by 8/3 ?  In other words, is it true that every graph of circular chromatic number 8/3 contains subgraph of circular chromatic number 5/2 ?

It is known [HZ02] if n is an integer and n < p/q, then n is unavoidable by p/q.

It is proved in that there are graph