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.Comment: 20 pages, 2 figures, 2 pages appendi

Two conjectures in Ramsey-Turan theory

Given graphs H 1 ,...,H k , a graph G is ( H 1 ,...,H k )-free if there is a k -edge-colouring φ : E ( G ) → [ k ] with no monochromatic copy of H i with edges of colour i for each i ∈ [ k ]. Fix a function f ( n ), the Ramsey-Tur ́an function RT( n,H 1 ,...,H k ,f ( n )) is the maximum number of edges in an n -vertex ( H 1 ,...,H k )-free graph with independence number at most f ( n ). We determine RT( n,K 3 ,K s ,δn ) for s ∈ { 3 , 4 , 5 } and sufficiently small δ , confirming a conjecture of Erd ̋os and S ́os from 1979. It is known that RT( n,K 8 ,f ( n )) has a phase transition at f ( n ) = Θ( √ n log n ). However, the value of RT( n,K 8 ,o ( √ n log n )) was not known. We determined this value by proving RT( n,K 8 ,o ( √ n log n )) = 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

On Ideal Secret-Sharing Schemes for kk-homogeneous access structures

A $k$-uniform hypergraph is a hypergraph where each $k$-hyperedge has exactly $k$ vertices. A $k$-homogeneous access structure is represented by a $k$-uniform hypergraph $\mathcal{H}$, in which the participants correspond to the vertices of hypergraph $\mathcal{H}$. A set of vertices can reconstruct the secret value from their shares if they are connected by a $k$-hyperedge, while a set of non-adjacent vertices does not obtain any information about the secret. One parameter for measuring the efficiency of a secret sharing scheme is the information rate, defined as the ratio between the length of the secret and the maximum length of the shares given to the participants. Secret sharing schemes with an information rate equal to one are called ideal secret sharing schemes. An access structure is considered ideal if an ideal secret sharing scheme can realize it. Characterizing ideal access structures is one of the important problems in secret sharing schemes. The characterization of ideal access structures has been studied by many authors~\cite{BD, CT,JZB, FP1,FP2,DS1,TD}. In this paper, we characterize ideal $k$-homogeneous access structures using the independent sequence method. In particular, we prove that the reduced access structure of $\Gamma$ is an $(k, n)$-threshold access structure when the optimal information rate of $\Gamma$ is larger than $\frac{k-1}{k}$, where $\Gamma$ is a $k$-homogeneous access structure satisfying specific criteria.Comment: 19 page