Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two conjectures in Ramsey-Tur\'an theory   Given graphs $H_1,\ldots, H_k$, a graph $G$ is $(H_1,\ldots, H_k)$-free if
there is a $k$-edge-colouring $\phi:E(G)\rightarrow [k]$ with no monochromatic
copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function
$f(n)$, the Ramsey-Tur\'an function $\textrm{RT}(n,H_1,\ldots,H_k,f(n))$ is the
maximum number of edges in an $n$-vertex $(H_1,\ldots,H_k)$-free graph with
independence number at most $f(n)$. We determine $\textrm{RT}(n,K_3,K_s,\delta
n)$ for $s\in\{3,4,5\}$ and sufficiently small $\delta$, confirming a
conjecture of Erd\H{o}s and S\'os from 1979. It is known that
$\textrm{RT}(n,K_8,f(n))$ has a phase transition at $f(n)=\Theta(\sqrt{n\log
n})$. However, the values of $\textrm{RT}(n,K_8, o(\sqrt{n\log n}))$ was not
known. We determined this value by proving $\textrm{RT}(n,K_8,o(\sqrt{n\log
n}))=\frac{n^2}{4}+o(n^2)$, answering a question of Balogh, Hu and Simonovits.
The proofs utilise, among others, dependent random choice and results from
graph packings.

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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th