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

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page

On Ramsey numbers of complete graphs with dropped stars

Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K\_n$, $n\geqslant N$, there is either a red copy of $G$ or a blue copy of $H$. Let $K\_n-K\_{1,s}$ be the complete graph on $n$ vertices from which the edges of $K\_{1,s}$ are dropped. In this note we present exact values for $r(K\_m-K\_{1,1},K\_n-K\_{1,s})$ and new upper bounds for $r(K\_m,K\_n-K\_{1,s})$ in numerous cases. We also present some results for the Ramsey number of Wheels versus $K\_n-K\_{1,s}$.Comment: 9 pages ; 1 table in Discrete Applied Mathematics, Elsevier, 201

Online Ramsey Numbers and the Subgraph Query Problem

The $(m,n)$-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red $K_m$ or a blue $K_n$ using as few turns as possible. The online Ramsey number $\tilde{r}(m,n)$ is the minimum number of edges Builder needs to guarantee a win in the $(m,n)$-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement $$\tilde{r}(n,n) \ge 2^{(2-\sqrt{2})n + O(1)}$$ for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement $$\tilde{r}(m,n) \ge n^{(2-\sqrt{2})m + O(1)}$$ for the off-diagonal case, where $m\ge 3$ is fixed and $n\rightarrow\infty$. Using a different randomized Painter strategy, we prove that $\tilde{r}(3,n)=\tilde{\Theta}(n^3)$, determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for $m \geq 4$. In connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph $H$ in a sufficiently large unknown Erd\H{o}s--R\'{e}nyi random graph $G(N,p)$ using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem.Comment: Corrected substantial error in the proof of Theorem

On two problems in Ramsey-Tur\'an theory

Alon, Balogh, Keevash and Sudakov proved that the $(k-1)$-partite Tur\'an graph maximizes the number of distinct $r$-edge-colorings with no monochromatic $K_k$ for all fixed $k$ and $r=2,3$, among all $n$-vertex graphs. In this paper, we determine this function asymptotically for $r=2$ among $n$-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with sub-linear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an $n$-vertex $K_k$-free graph $G$ with $\alpha(G)=o(n)$. The extremal graphs have similar structure to the extremal graphs for the classical Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page

A characterization of Ramsey graphs for R(3,4)

The Ramsey number R(ω, α) is the minimum number n such that every graph G with |V(G)| ≥ n has an induced subgraph that is isomorphic to a complete graph on ω vertices, Kω, or has an independent set of size α, Nα. Graphs having fewer than n vertices that have no induced subgraph isomorphic to K ω or Nα form a class of Ramsey graphs, denoted ℜ(ω, α). This dissertation establishes common structure among several classes of Ramsey graphs and establishes the complete list of ℜ(3, 4). The process used to find the complete list for ℜ(3, 4) can be extended to find other Ramsey numbers and Ramsey graphs. The technique for finding a complete list for ℜ(ω, α), a) is inductive on n vertices in that a complete list of all graphs in ℜ(ω, α) having exactly n vertices can be used to find the complete list n + 1 vertices. This process can be repeated until any extension is not in ℜ(ω, α), and thus R(ω, α) has been determined. We conclude by showing how to extend methods presented in proving R(3, 4) in finding R(5, 5)