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We construct an edge-coloring of $K_{N}$ (for $N = 3432$) in colors red, dark blue, and light blue, such that there are no monochromatic blue triangles and such that the coloring satisfies a certain strong universal-existential property.…

Given integers $2\le t \le k+1 \le n$, let $g_k(t,n)$ be the minimum $N$ such that every red/blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ yields either a $(k+1)$-set containing $t$ red $k$-subsets, or an $n$-set with all of its…

Kneser's 1955 conjecture -- proven by Lov\'asz in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to…

Given a graph $G$, an exact $r$-coloring of $G$ is a surjective function $c:V(G) \to [1,\dots,r]$. An arithmetic progression in $G$ of length $j$ with common difference $d$ is a set of vertices $\{v_1,\dots, v_j\}$ such that…

We show that $\sqrt{k}\cdot r^{k/2}$ is a threshold interval length where, under mild conditions, almost every $r$-coloring of an interval of longer length contains a monochromatic $k$-term arithmetic progression, while almost no…

The chromatic number of the random graph $\mathcal{G}(n,p)$ has long been studied and has inspired several landmark results. In the case where $p = d/n$, Achlioptas and Naor showed the chromatic number is asymptotically concentrated at…

For many random Constraint Satisfaction Problems, by now, we have asymptotically tight estimates of the largest constraint density for which they have solutions. At the same time, all known polynomial-time algorithms for many of these…

A Kneser graph $KG_{n,k}$ is a graph whose vertices are in one-to-one correspondence with $k$-element subsets of $[n],$ with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lov\'asz…

The dichromatic number of a digraph $D$ is the smallest $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs, and the dichromatic number of an undirected graph is the maximum dichromatic number over all its orientations.…

A 2-coloring of a hypergraph is a mapping from its vertices to a set of two colors such that no edge is monochromatic. Let $H_k(n,m)$ be a random $k$-uniform hypergraph on $n$ vertices formed by picking $m$ edges uniformly, independently…

A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…

We wish to generate list-decodable codes over small alphabets using as little randomness as possible. Specifically, we hope to generate codes achieving what we term the Elias bound, which means that they are $(\rho,L)$-list-decodable with…

Hadwiger's Conjecture asserts that every $K_h$-minor-free graph is properly $(h-1)$-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with…

We consider the problem of constructing deletion correcting codes over a binary alphabet and take a graph theoretic view. An $n$-bit $s$-deletion correcting code is an independent set in a particular graph. We propose constructing such a…

We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \colon \{0,1\}^k \rightarrow \{0,1\}^n$. Specifically, we prove: (1) If $C$ is a linear design 3-LCC, then $n \geq 2^{(1 - o(1))\sqrt{k} }$. A design…

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges.…

The dispersion of a point set $P\subset[0,1]^d$ is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect $P$. Here, we show a construction of low-dispersion point sets, which can be deduced from…

In 1961 Erd\H{o}s and Hajnal introduced the quantity $m(n)$ as the minimum number of edges in an $n$-uniform hypergraph with chromatic number at least 3. The best known lower and upper bounds for $ m(n) $ are $ c_1 \sqrt{\frac{n}{\ln n}}…

In an undirected graph, a proper (k,i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k,i)-coloring problem is to compute the minimum number of colors…

The chromatic number $\chi(\mathbb{R}^n)$ of the Euclidean space $\mathbb{R}^n$ is the smallest number of colors sufficient for coloring all points of the space in such a way that any two points at the distance 1 have different colors. In…