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The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,\dots,a_n\in \mathbb{R}^d$, any $x\in \mathbb{R}^d$, and uniformly random $(\xi_1,\dots,\xi_n)\in\{-1,1\}^n$, we have…

Erd\H{o}s, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285] conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$ and minimum degree $\delta\geq 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot…

Given an integer $r\gs 2$, an important theorem first proved by B. Andr\'asfai, P. Erd\H{o}s, and V. T. S\'os states that any $K_{r+1}$--free graph on $n$ vertices whose minimum degree is greater than $(3r-4)n/(3r-1)$ is $r$--colourable,…

Given graphs $G, H$ and an integer $q \ge 2$, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$ such that every copy of $H$ receives at least $q$ colours. In this paper, we prove…

For a non-decreasing sequence $S=(s_1,s_2,\ldots)$ of positive integers, a partition of the vertex set of a graph $G$ into subsets $X_1,\ldots, X_\ell$, such that vertices in $X_i$ are pairwise at distance greater than $s_i$ for every…

Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey…

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $w_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\chi:[1,w_{\mathrm{\mathfrak{z}}}(k;r)] \rightarrow \{0,1,\dots,r-1\}$ admits a $k$-term…

The following question was asked by Prendiville: given an $r$-colouring of the interval $\{2, \dotsc, N\}$, what is the minimum number of monochromatic solutions of the equation $xy = z$? For $r=2$, we show that there are always…

We say a pair of integers $(a, b)$ is findable if the following is true. For any $\delta > 0$ there exists a $p_0$ such that for any prime $p \ge p_0$ and any red-blue colouring of $\mathbb{Z} /p\mathbb{Z}$ in which each colour has density…

Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $\chi_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in…

Lehel conjectured that in every $2$-coloring of the edges of $K_n$, there is a vertex disjoint red and blue cycle which span $V(K_n)$. \L uczak, R\"odl, and Szemer\'edi proved Lehel's conjecture for large $n$, Allen gave a different proof…

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors.…

Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different…

The Ramsey number $R(G_1,\dots,G_k)$ is the smallest $n$ such that every $k$-coloring of the edges of $K_n$ contains a monochromatic copy of $G_i$ in color $i$. Ramsey numbers are challenging to compute, and few are known exactly. We use…

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n),…

For given simple graphs $G_1, G_2, \ldots , G_t$, the Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that if the edges of the complete graph $K_n$ are partitioned into $t$ disjoint color classes giving…

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…

When the sequences of squares of primes is coloured with $K$ colours, where $K \geq 1$ is an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as a sum of no more than $s(K)$ squares of…

For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic…

The two-dimensional signed small ball inequality states that for all possible choices of signs, $$ \left\| \sum_{|R| = 2^{-n}}{ \varepsilon_R h_R} \right\|_{L^{\infty}} \gtrsim n,$$ where the summation runs over all dyadic rectangles in the…