arXiv subjects
Publications and source records attributed to József Balogh.
Let $r \ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Győri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Turán graph on $n$ vertices and $r - 1$ parts, then $G$ has at least $(2 - o(1))k/r$ edge disjoint $r$-cliques. We prove this conjecture.
Given a graph $H$, the Ramsey number $R(H)$ is the smallest positive integer $n$ such that every $2$-edge-colouring of $K_n$ yields a monochromatic copy of $H$. We write $mH$ to denote the union of $m$ vertex-disjoint copies of $H$. The members of the family $\{mH:m\ge1\}$ are also known as $H$-tilings. A well-known result of Burr, Erdős and Spencer states that $R(mK_3)=5m$ for every $m\ge2$. On the other hand, Moon proved that every $2$-edge-colouring of $K_{3m+2}$ yields a $K_3$-tiling consisting of $m$ monochromatic copies of $K_3$, for every $m\ge2$. Crucially, in Moon's result, distinct copies of $K_3$ might receive different colours. In this paper, we investigate the analogous questions where the complete host graph is replaced by a graph of large minimum degree. We determine the (asymptotic) minimum degree threshold for forcing a~$K_3$-tiling covering a prescribed proportion of the vertices in a $2$-edge-coloured graph such that every copy of $K_3$ in the tiling is monochromatic. We also determine the largest size of a monochromatic $K_3$-tiling one can guarantee in any $2$-edge-coloured graph of large minimum degree. These results therefore provide generalisations of the theorems of Moon and Burr-Erdős-Spencer to the setting of dense graphs. It is also natural to consider generalisations of these problems to $r$-edge-colourings (for $r \geq 2$) and for $H$-tilings (for arbitrary graphs $H$). We prove some results in this direction and propose several open questions.
Let $B(2d-1, d)$ be the subgraph of the hypercube $\mathcal{Q}_{2d-1}$ induced by its two largest layers. Duffus, Frankl and Rödl proposed the problem of finding the asymptotics for the logarithm of the number of maximal independent sets in $B(2d-1, d)$. Ilinca and Kahn determined the logarithmic asymptotics and reiterated the question of what their order of magnitude is. We show that the number of maximal independent sets in $B(2d-1,d)$ is \[ \left(1+o(1)\right)(2d-1)\exp\left(\frac{(d-1)^2}{2^{2d-1}}\binom{2d-2}{d-1}\right)\cdot 2^{\binom{2d-2}{d-1}}, \] and describe their typical structure. The proof uses a new variation of Sapozhenko's Graph Container Lemma, a new isoperimetric lemma, a theorem of Hujter and Tuza on the number of maximal independent sets in triangle-free graphs and a stability version of their result by Kahn and Park, among other tools.
We estimate the maximum possible number of cliques of size $r$ in an $n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2, \ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on the Turán number of the complete $r$-partite hypergraphs gives the upper bound $O\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. We improve this to $o\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions.
There are various different notions measuring extremality of hypergraphs. In this survey we compare the recently introduced notion of the codegree squared extremal function with the Turán function, the minimum codegree threshold and the uniform Turán density. The codegree squared sum $\textrm{co}_2(G)$ of a $3$-uniform hypergraph $G$ is defined to be the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $x,y$. In other words, this is the square of the $\ell_2$-norm of the codegree vector. We are interested in how large $\textrm{co}_2(G)$ can be if we require $G$ to be $H$-free for some $3$-uniform hypergraph $H$. This maximum value of $\textrm{co}_2(G)$ over all $H$-free $n$-vertex $3$-uniform hypergraphs $G$ is called the codegree squared extremal function, which we denote by $\textrm{exco}_2(n,H)$. We systemically study the extremal codegree squared sum of various $3$-uniform hypergraphs using various proof techniques. Some of our proofs rely on the flag algebra method while others use more classical tools such as the stability method. In particular, we (asymptotically) determine the codegree squared extremal numbers of matchings, stars, paths, cycles, and $F_5$, the $5$-vertex hypergraph with edge set $\{123,124,345\}$. Additionally, our paper has a survey format, as we state several conjectures and give an overview of Turán densities, minimum codegree thresholds and codegree squared extremal numbers of popular hypergraphs. We intend to update the arXiv version of this paper regularly.
We say a finite poset $P$ is a tree poset if its Hasse diagram is a tree. Let $k$ be the length of the largest chain contained in $P$. We show that when $P$ is a fixed tree poset, the number of $P$-free set systems in $2^{[n]}$ is $2^{(1+o(1))(k-1){n \choose \lfloor n/2\rfloor}}$. The proof uses a generalization of a theorem by Boris Bukh together with a variation of the multiphase graph container algorithm.
The minimum positive co-degree of a nonempty $r$-graph $H$, denoted by $δ_{r-1}^+(H)$, is the largest integer $k$ such that for every $(r-1)$-set $S \subset V(H)$, if $S$ is contained in a hyperedge of $H$, then $S$ is contained in at least $k$ hyperedges of $H$. Given a family $\mathcal{F}$ of $r$-graphs, the positive co-degree Turán function $\mathrm{co^+ex}(n,\mathcal{F})$ is the maximum of $δ_{r-1}^+(H)$ over all $n$-vertex $r$-graphs $H$ containing no member of $\mathcal{F}$. The positive co-degree density of $\mathcal{F}$ is $γ^+(\mathcal{F}) = \underset{n \rightarrow \infty}{\lim} \frac{\mathrm{co^+ex}(n,\mathcal{F})}{n}.$ While the existence of $γ^+(\mathcal{F})$ is proved for all families $\mathcal{F}$, only few positive co-degree densities are known exactly. For a fixed $r \geq 2$, we call $α\in [0,1]$ an achievable value if there exists a family of $r$-graphs $\mathcal{F}$ with $γ^+(\mathcal{F}) = α$, and call $α$ a jump if for some $δ> 0$, there is no family $\mathcal{F}$ with $γ^+(\mathcal{F}) \in (α, α+ δ)$. Halfpap, Lemons, and Palmer showed that every $α\in [0, \frac{1}{r})$ is a jump. We extend this result by showing that every $α\in [0, \frac{2}{2r -1})$ is a jump. We also show that for $r = 3$, the set of achievable values is infinite, more precisely, $\frac{k-2}{2k-3}$ for every $k \geq 4$ is achievable. Finally, we determine two additional achievable values for $r=3$ using flag algebra calculations.
A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n\geq 2k + c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\geq 2k+ 100\ln k$. Our proof uses, among others, Sapozhenko's graph container lemma and the Das-Tran removal lemma.
For graphs $F$ and $H$, let $f_{F,H}(n)$ be the minimum possible size of a maximum $F$-free induced subgraph in an $n$-vertex $H$-free graph. This notion generalizes the Ramsey function and the Erdős--Rogers function. Establishing a container lemma for the $F$-free subgraphs, we give a general upper bound on $f_{F,H}(n)$, assuming the existence of certain locally dense $H$-free graphs. In particular, we prove that for every graph $F$ with $\mathrm{ex}(m,F) = O(m^{1+α})$, where $α\in [0,1/2)$, we have \[ f_{F, K_3}(n) = O\left(n^{\frac{1}{2-α}}\left(\log n\right)^{\frac{3}{2- α}}\right) \quad \textrm{and} \quad f_{F, K_4}(n) = O\left(n^{\frac{1}{3-2α}}\left(\log n\right)^{\frac{6}{3-2α}}\right). \] For the cases where $F$ is a complete multipartite graph, letting $s = \sum_{i=1}^r s_i$, we prove that \[ f_{K_{s_1,\ldots,s_r}, K_{r+2}}(n) = O \left( n^{\frac{2s -3}{4s -5}} (\log n)^{3} \right). \] We also make an observation which improves the bounds of $\mathrm{ex}(G(n,p),C_4)$ by a polylogarithmic factor.
We investigate a random geometric graph model introduced by Bonato and Janssen. The vertices are the points of a countable dense set $S$ in a (necessarily separable) normed vector space $X$, and each pair of points are joined independently with some fixed probability $p$ (with $0<p<1$) if they are less than distance $1$ apart. A countable dense set $S$ in a normed space is Rado, if the resulting graph is almost surely unique up to isomorphism: that is any two such graphs are, almost surely, isomorphic. Not surprisingly, understanding which sets are Rado is closely related to the geometry of the underlying normed space. It turns out that a key question is in which spaces must step-isometries (maps that preserve the integer parts of distances) on dense subsets necessarily be isometries. We answer this question for a large class of Banach spaces including all strictly convex reflexive spaces. In the process we prove results on the interplay between the norm topology and weak topology that may be of independent interest. As a consequence of these Banach space results we show that almost all countable dense sets in strictly convex reflexive spaces are strongly non-Rado (that is, any two graphs are almost surely non-isomorphic). However, we show that there do exist Rado sets even in $\ell_2$. Finally we construct a Banach spaces in which all countable dense set are strongly non-Rado.
Let $X$ be an $n$-element point set in the $k$-dimensional unit cube $[0,1]^k$ where $k \geq 2$. According to an old result of Bollobás and Meir (1992), there exists a cycle (tour) $x_1, x_2, \ldots, x_n$ through the $n$ points, such that $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} \leq c_k$, where $|x-y|$ is the Euclidean distance between $x$ and $y$, and $c_k$ is an absolute constant that depends only on $k$, where $x_{n+1} \equiv x_1$. From the other direction, for every $k \geq 2$ and $n \geq 2$, there exist $n$ points in $[0,1]^k$, such that their shortest tour satisfies $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} = 2^{1/k} \cdot \sqrt{k}$. For the plane, the best constant is $c_2=2$ and this is the only exact value known. Bollob{á}s and Meir showed that one can take $c_k = 9 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ for every $k \geq 3$ and conjectured that the best constant is $c_k = 2^{1/k} \cdot \sqrt{k}$, for every $k \geq 2$. Here we significantly improve the upper bound and show that one can take $c_k = 3 \sqrt5 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ or $c_k = 2.91 \sqrt{k} \ (1+o_k(1))$. Our bounds are constructive. We also show that $c_3 \geq 2^{7/6}$, which disproves the conjecture for $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás--Meir conjecture is proposed.
We address a problem which is a generalization of Turán-type problems recently introduced by Imolay, Karl, Nagy and Váli. Let $F$ be a fixed graph and let $G$ be the union of $k$ edge-disjoint copies of $F$, namely $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$, where each $F_i$ is isomorphic to a fixed graph $F$ and $E(F_i)\cap E(F_j)=\emptyset$ for all $i \neq j$. We call a subgraph $H\subseteq G$ multicolored if $H$ and $F_i$ share at most one edge for all $i$. Define $\text{ex}_F(H,n)$ to be the maximum value $k$ such that there exists $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$ on $n$ vertices without a multicolored copy of $H$. We show that $\text{ex}_{C_5}(C_3,n) \le n^2/25 + 3n/25+o(n)$ and that all extremal graphs are close to a blow-up of the 5-cycle. This bound is tight up to the linear error term.
The Ramsey-Turán problem for $K_p$ asks for the maximum number of edges in an $n$-vertex $K_p$-free graph with independence number $o(n)$. In a natural generalization of the problem, cliques larger than the edge $K_2$ are counted. Let {\bf RT}$(n,\#K_q,K_p,o(n))$ denote the maximum number of copies of $K_q$ in an $n$-vertex $K_p$-free graph with independence number $o(n)$. Balogh, Liu and Sharifzadeh determined the asymptotics of {\bf RT}$(n,\# K_3,K_p,o(n))$. In this paper we will establish the asymptotics for counting copies of $K_4$, $K_5$, and for the case $p \geq 5q$. We also provide a family of counterexamples to a conjecture of Balogh, Liu and Sharifzadeh.
We extend the notion of proper elements to all Coxeter groups. For all infinite families of finite Coxeter groups we prove that the probability a random element is proper goes to zero in the limit. This proves a conjecture of the third author and A. Yong regarding the proportion of Schubert varieties that are Levi spherical for all infinite families of Weyl groups. We also enumerate the proper elements in the exceptional Coxeter groups.
Denote by $\mathcal{C}^-_{\ell}$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\ell$ vertices. It is conjectured that the Turán density of $\mathcal{C}^-_{5}$ is $1/4$. In this paper, we make progress toward this conjecture by proving that the Turán density of $\mathcal{C}^-_{\ell}$ is $1/4$, for every sufficiently large $\ell$ not divisible by $3$. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament. A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidický.
A result of Balogh, Csaba, Jing and Pluhár yields the minimum degree threshold that ensures a $2$-coloured graph contains a perfect matching of significant colour-bias (i.e., a perfect matching that contains significantly more than half of its edges in one colour). In this note we prove an analogous result for perfect matchings in $k$-uniform hypergraphs. More precisely, for each $2\leq \ell <k$ and $r\geq 2$ we determine the minimum $\ell$-degree threshold for forcing a perfect matching of significant colour-bias in an $r$-coloured $k$-uniform hypergraph.
In the Constructor-Blocker game, two players, Constructor and Blocker, alternatively claim unclaimed edges of the complete graph $K_n$. For given graphs $F$ and $H$, Constructor can only claim edges that leave her graph $F$-free, while Blocker has no restrictions. Constructor's goal is to build as many copies of $H$ as she can, while Blocker attempts to stop this. The game ends once there are no more edges that Constructor can claim. The score $g(n,H,F)$ of the game is the number of copies of $H$ in Constructor's graph at the end of the game, when both players play optimally and Constructor plays first. In this paper, we extend results of Patkós, Stojaković and Vizer on $g(n, H, F)$ to many pairs of $H$ and $F$: We determine $g(n, H, F)$ when $H=K_r$ and $χ(F)>r$, also when both $H$ and $F$ are odd cycles, using Szemerédi's Regularity Lemma. We also obtain bounds of $g(n, H, F)$ when $H=K_3$ and $F=K_{2,2}$.
In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $α>0$, there exists a constant $C$ such that for every $n$-vertex digraph of minimum semi-degree at least $αn$, if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree $1$. Our proofs make use of a variant of an absorbing method of Montgomery.