Chi-boundedness and Upperbounds on Chromatic Number I am new to graph theory and am a bit confused about the difference b/w the chi-boundedness and upper-bounds on chromatic number. Can we say that chi-boundedness is a form of an upperbound of the chromatic number? Could someone help me to clarify?
Definitions

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*Chromatic Number: Minimal number of colours needed to colour the vertices in such a way that no two adjacent vertices have the same colour

*Chi-boundedness: Ideal(I) is chi-bounded if there is a func f, called chi-bounding func, s.t. $\chi$(G) < f($\omega$(G)) for all G in I

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*I understand that chi-boundedness basically considers the following question: If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs?



 A: The notion of $\chi$-boundedness is a generalization of perfect graphs. A graph $G$ is perfect if $\chi(H)=\omega(H)$ for each induced subgraph $H$ of $G$.
A graph $G$ is $\chi$-bounded if there exists a function $f:\mathbb{N}\to \mathbb{N}$ such that $\chi(H) \leq f(\omega(H))$ for each induced subgraph $H$ of $G$.
If we let $f:\omega \mapsto \omega$ (the identity function), then $G$ is a perfect graph iff $G$ is $\chi$-bounded by $f$. Hence, the above is a direct generalization.
In that sense, a $\chi$-binding function is an upper bound for the chromatic number, but in form of a function from its clique number $\omega$ to a natural number.
For example, the class of even-hole-free graphs, that is the class of graphs that forbidds induced cycles of even length, is $\chi$-bounded with $\chi$-binding function $\chi(G) \leq 2\omega(G)-1$ as shown by Addario-Berry et.al.
Question: Are all graphs $\chi$-bounded by some function $f$? No. There are graphs that are triangle free, thus clique number strictly smaller than $3$, but with an arbitrary large chromatic number (consider the Mycielskian construction for an example).
Answering the question whether or not a given family of graphs is $\chi$-bounded or not is far from trivial and an open research question for many classes of graphs.
There is another interesting problem.
Problem: Given a class of graphs that is $\chi$-bounded by some function $f$. Is $f$ the smallest $\chi$-binding function for this class?
Answering the above is an open research question for many graph classes.
Your second questions asks what we can say about the induced subgraphs of a graph $G$ if we are provided with a $\chi$-binding function. For a concrete graph, these are the usual answers one could provide if we know the chromatic number of a graph  (which is the same as saying we know the $\chi$-binding function of a specific graph that provides a tight bound). This is not interesting. It is more interesting to consider a whole family of graphs and its $\chi$-binding function, because it allows us to make certain statements about the structure of the whole family.
Is there a general answer about the structure of a class of graphs if we know its $\chi$-binding function? I am afraid that there is none, but I can't say this for sure.
If you want to dive deeper into this topic, I suggest the following two survey papers:
[1] "Schiermeyer, Ingo, and Bert Randerath. "Polynomial $\chi$-Binding Functions and Forbidden Induced Subgraphs: A Survey." Graphs and Combinatorics 35.1 (2019): 1-31."
[2] "Scott, A., & Seymour, P. (2020). A survey of χ-boundedness." (I have only found it on arXiv)
