Related papers: Hypergraph Removal Lemmas via Robust Sharp Thresho…

Let H be a 3-uniform hypergraph of order n with clique number k such that the intersection of all maximum cliques of H is empty. For fixed m=n-k, Szemer\'edi and Petruska conjectured the sharp bound $n\leq {m+2\choose 2}$. In this note the…

We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated $(p,q)$-theorem in discrete geometry. In particular, for a graph $G$ with bounded clique number and a natural…

We introduce a regularity method for sparse graphs, with new regularity and counting lemmas which use the Schatten-von-Neumann norms to measure uniformity. This leads to $k$-cycle removal lemmas in subgraphs of mildly-pseudorandom graphs,…

Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the…

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…

An old and well-known conjecture of Frankl and F\"{u}redi states that the Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite…

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists $M$ such that every triangle-free…

Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a…

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…

Szemer\'edi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's regularity lemma…

The problem of detecting edge correlation between two Erd\H{o}s-R\'enyi random graphs on $n$ unlabeled nodes can be formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are sampled independently; under the…

For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and…

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]$ with $m$ edges, whenever $n$ and the nullity $m-n+1$ tend to infinity. Asymptotic formulae for the number of connected $r$-uniform…

Let $A$ be a subset of positive relative upper density of $\PP^d$, the $d$-tuples of primes. We prove that $A$ contains an affine copy of any finite set $F\subs\Z^d$, which provides a natural multi-dimensional extension of the theorem of…

The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type which asks, for fixed integers ${\ell}$ and…

In 1975 Lov\'{a}sz conjectured that every $r$-partite, $r$-uniform hypergraph contains $r-1$ vertices whose deletion reduces the matching number. If true, this statement would imply a well-known conjecture of Ryser from 1971, which states…

Let a_1,...,a_k satisfy a_1+...+a_k=1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting…

Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set $[N]=\{1,2,\hdots,N\}$ was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph…

We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices…

The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the…