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Domagoj Bradač

Publications and source records attributed to Domagoj Bradač.

At least 19 recordsLinked to original sources
arXiv2026

On the asymptotics of the Erdős-Rogers function

The Erdős-Rogers function $f_{\ell,s}(n)$ is the largest order of a $K_\ell$-free induced subgraph guaranteed to exist in every $K_s$-free graph on $n$ vertices. While this function is well understood for $s=\ell+1$, the case where $s$ is much larger than $\ell$ has remained wide open. A long-standing lower bound of Sudakov states that $f_{\ell,s}(n)\geq n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$, while a recent result of Bradač shows that $f_{\ell,s}(n)\leq n^{\frac{\ell-1}{s-1}+o(1)}$. In this paper, we close this gap asymptotically by proving that $f_{\ell,s}(n)= n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$. More precisely, we prove that for all $2\leq \ell<s$, we have $f_{\ell,s}(n)\leq n^{\frac{\ell}{2s-\ell}+o(1)}$. Our proof builds on Bradač's recent tight construction for off-diagonal Ramsey numbers, which can be viewed as the $\ell=2$ case of our result.

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arXiv2026

Off-diagonal Ramsey numbers

For positive integers $s$ and $k$, the Ramsey number $r(s,k)$ is the minimum integer $n$ such that any graph on $n$ vertices contains a clique of size $s$ or an independent set of size $k$. We prove that for any fixed $s \ge 3$ and $k$ tending to infinity, the off-diagonal Ramsey numbers satisfy \[ r(s, k) \ge Ω\left(\frac{k^{s-1}}{(\log k)^{2s-4}} \right), \] which matches, up to polylogarithmic factors, the upper bound established over 90 years ago by Erdős and Szekeres. For $s \ge 5,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen polylogarithmic improvements.

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arXiv2026

Hamiltonicity of regular sublinear expanders

We say that a $d$-regular graph is a $γ$-expander if for every not too large set of vertices $S$, there are at least $γd |S|$ edges leaving $S$, and we say that a graph $G$ is $γ$-far from bipartite if at least $γe(G)$ edges need to be removed to make it bipartite. We prove that there exists an absolute constant $K$ such that any $n$-vertex $d$-regular $γ$-expander with $d \ge (γ^{-1} \log n)^K$ is Hamiltonian, provided that it is bipartite or $γ$-far from bipartite. As applications, we obtain highly robust versions of recent important results on the Hamiltonicity of Cayley graphs and Kneser graphs. As part of our proof, we prove a random connecting lemma for sublinear expanders which might be of independent interest.

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arXiv2026

Coloring small locally sparse degenerate graphs and related problems

The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and Zykov from the early 50s show that there exist $d$-degenerate $(d+1)$-chromatic graphs that are triangle-free, however these constructions grow rapidly with $d$. Motivated by this and addressing a problem posed by the second author at the Oberwolfach Graph Theory workshop, we prove that the minimum order $f(d)$ of a $d$-degenerate triangle-free graph of chromatic number $d+1$ satisfies $e^{Ω(d)}\le f(d)\le e^{O(d^2\log d)}.$ The lower bound follows from a novel upper bound on the chromatic number of triangle-free graphs: Every triangle-free $d$-degenerate graph $G$ on $n \le e^{O(d)}$ vertices satisfies $$χ(G)\le O\left(\frac{d}{\log\left(d/\log n\right)}\right).$$ We extend this to a more general result about degenerate graphs with sparse neighborhoods, which has applications to many graph coloring problems: For example, we prove that every counterexample to Hadwiger's conjecture with parameter $t$ must have a complete bipartite subgraph with one exponentially large side ($K_{a,b}$ where $a=(\log t)^{1/2-o(1)}$ and $b=e^{t^{1-o(1)}}$) or a small and very dense subgraph (of order $\le t$ with $t^{2-o(1)}$ edges) in some neighborhood. For the upper bound on $f(d)$ we establish a surprising connection between $f(d)$ and the on-line-chromatic number $g(n)$ of $n$-vertex triangle-free graphs. We also give an asymptotic improvement of the previous best upper bound for $g(n)$ due to Lovász, Saks and Trotter from 1989. Along the way we disprove a generalization of Harris' fractional coloring conjecture to graphs of bounded clique number and raise numerous problems which open up interesting directions to explore for future research.

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arXiv2025

Ramsey numbers of digraphs with local edge structure

One of the classical topics in graph Ramsey theory is the study of which $n$-vertex graphs have Ramsey numbers that are linear in $n$. In this paper, we consider this problem in the context of directed graphs. The oriented Ramsey number of a digraph $G$ is the smallest integer $N$ such that every $N$-vertex tournament contains a copy of $G$. We prove that every bounded-degree acyclic digraph with a ``local edge structure'' has a linear oriented Ramsey number. More precisely, we say that a digraph $G$ has graded bandwidth $w$ if its vertices can be partitioned into sets $V_1, \dots, V_H$ such that all edges $uv \in E(G)$ with $u \in V_i$ and $v \in V_j$ satisfy $1 \leq j - i \leq w$. We prove that $\vec{r}(G) \leq 3^{57Δw} |V(G)|$ for any acyclic $G$ with graded bandwidth $w$ and maximum degree $Δ$. This provides a common generalization of several prior results, including on digraphs of bounded height, of digraphs of bounded bandwidth, and blowups of bounded-degree oriented trees. This notion also captures a wide variety of natural digraphs, such as oriented grids and hypercubes.

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arXiv2025

Lower bounds for Ramsey numbers of bounded degree hypergraphs

We prove that, for all $k \ge 3,$ and any integers $Δ, n$ with $n \ge Δ,$ there exists a $k$-uniform hypergraph on $n$ vertices with maximum degree at most $Δ$ whose $4$-color Ramsey number is at least $\mathrm{tw}_k(c_k Δ) \cdot n$, for some constant $c_k > 0$, where $\mathrm{tw}_k$ denotes the tower function. For $k \ge 4,$ this is tight up to the constant $c_k$ and for $k = 3$ it is known to be tight up to a factor of $\log Δ$ on top of the tower. It extends a well-known result of Graham, Rödl and Ruciński for graphs and answers a question of Conlon, Fox and Sudakov from 2008.

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arXiv2024

Ordered Ramsey numbers of graphs with $m$ edges

Given a vertex-ordered graph $G$, the ordered Ramsey number $r_<(G)$ is the minimum integer $N$ such that every $2$-coloring of the edges of the complete ordered graph $K_N$ contains a monochromatic ordered copy of $G$. Motivated by a similar question posed by Erdős and Graham in the unordered setting, we study the problem of bounding the ordered Ramsey number of any ordered graph $G$ with $m$ edges and no isolated vertices. We prove that $r_<(G) \leq e^{10^9 \sqrt{m} (\log \log m)^{3/2}}$ for any such $G$, which is tight up to the $(\log \log m)^{3/2}$ factor in the exponent. As a corollary, we obtain the corresponding bound for the oriented Ramsey number of a directed graph with $m$ edges.

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arXiv2024

Unique subgraphs are rare

A folklore result attributed to Pólya states that there are $(1 + o(1))2^{\binom{n}{2}}/n!$ non-isomorphic graphs on $n$ vertices. Given two graphs $G$ and $H$, we say that $G$ is a unique subgraph of $H$ if $H$ contains exactly one subgraph isomorphic to $G$. For an $n$-vertex graph $H$, let $f(H)$ be the number of non-isomorphic unique subgraphs of $H$ divided by $2^{\binom{n}{2}}/n!$ and let $f(n)$ denote the maximum of $f(H)$ over all graphs $H$ on $n$ vertices. In 1975, Erdős asked whether there exists $δ>0$ such that $f(n)>δ$ for all $n$ and offered $\$100$ for a proof and $\$25$ for a disproof, indicating he does not believe this to be true. We verify Erdős' intuition by showing that $f(n)\rightarrow 0$ as $n$ tends to infinity, i.e. no graph on $n$ vertices contains a constant proportion of all graphs on $n$ vertices as unique subgraphs.

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arXiv2024

Clique density vs blowups

A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov's result in the following form. Given $r,t\in\mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include: For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline{G}$ has positive triangle density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline{G}$ has positive $K_r$-density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2. Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $Ω(\frac{n}{\log{n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $Ω(n)$, where the constant $(2h-2)^{r}+1$ is optimal. The $\frac{n}{\log n}$ size of the blowups in all our results are optimal up to a constant factor.

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arXiv2024

Counting subgraphs in locally dense graphs

A graph $G$ is said to be $p$-locally dense if every induced subgraph of $G$ with linearly many vertices has edge density at least $p$. A famous conjecture of Kohayakawa, Nagle, Rödl, and Schacht predicts that locally dense graphs have, asymptotically, at least as many copies of any fixed graph $H$ as are found in a random graph of edge density $p$. In this paper, we prove several results around the KNRS conjecture. First, we prove that certain natural gluing operations on $H$ preserve this property, thus proving the conjecture for many graphs $H$ for which it was previously unknown. Secondly, we study a stability version of this conjecture, and prove that for many graphs $H$, approximate equality is attained in the KNRS conjecture if and only if the host graph $G$ is quasirandom. Finally, we introduce a weakening of the KNRS conjecture, which requires the host graph to be nearly degree-regular, and prove this conjecture for a larger family of graphs. Our techniques reveal a surprising connection between these questions, semidefinite optimization, and the study of copositive matrices.

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arXiv2024

The growth rate of multicolor Ramsey numbers of $3$-graphs

The $q$-color Ramsey number of a $k$-uniform hypergraph $G,$ denoted $r(G;q)$, is the minimum integer $N$ such that any coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $G$. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behaviour of $r(G;q)$ for fixed $G$ and $q$ tending to infinity. In this paper we study this problem for $3$-uniform hypergraphs and determine the tower height of $r(G;q)$ as a function of $q$. More precisely, given a hypergraph $G$, we determine when $r(G; q)$ behaves polynomially, exponentially or double-exponentially in $q$. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.

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arXiv2023

The extremal number of cycles with all diagonals

In 1975, Erdős asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erdős observed that the upper bound $O(n^{5/3})$ holds since the complete bipartite graph $K_{3,3}$ can be viewed as a cycle of length six with all diagonals. In this paper, we resolve this old problem. We prove that there exists a constant $C$ such that every $n$-vertex with $Cn^{3/2}$ edges contains a cycle with all diagonals. Since any cycle with all diagonals contains cycles of length four, this bound is best possible using well-known constructions of graphs without a four-cycle based on finite geometry. Among other ideas, our proof involves a novel lemma about finding an `almost-spanning' robust expander which might be of independent interest.

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arXiv2023

Ramsey numbers of hypergraphs of a given size

The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erdős and Graham asked to maximize the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible $q$-color Ramsey number of a $k$-uniform hypergraph with $m$ edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where $\mathrm{tw}$ denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs.

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arXiv2023

Effective bounds for induced size-Ramsey numbers of cycles

The induced size-Ramsey number $\hat{r}_\text{ind}^k(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have such that for any $k$-coloring of its edges, there exists a monochromatic copy of $H$ which is an induced subgraph of $G$. In 1995, in their seminal paper, Haxell, Kohayakawa and Luczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., $\hat{r}_\text{ind}^k(C_n)\leq Cn$ for some $C=C(k)$. The constant $C$ comes from the use of the regularity lemma, and has a tower type dependence on $k$. In this paper we significantly improve these bounds, showing that $\hat{r}_\text{ind}^k(C_n)\leq O(k^{102})n$ when $n$ is even, thus obtaining only a polynomial dependence of $C$ on $k$. We also prove $\hat{r}_\text{ind}^k(C_n)\leq e^{O(k\log k)}n$ for odd $n$, which almost matches the lower bound of $e^{Ω(k)}n$. Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies $\hat{r}^k(C_n)=e^{O(k)}n$ for odd $n$. This substantially improves the best previous result of $e^{O(k^2)}n$, and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.

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arXiv2023

Large cliques or co-cliques in hypergraphs with forbidden order-size pairs

The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $ε>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^ε$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every $S \subseteq \{0,1,2,3,4\}$, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. In most cases the bounds are essentially tight. We also determine, for all $S$, whether the growth rate is polynomial or polylogarithmic. Some open problems remain.

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arXiv2023

On Ramsey size-linear graphs and related questions

In this paper we prove several results on Ramsey numbers $R(H,F)$ for a fixed graph $H$ and a large graph $F$, in particular for $F = K_n$. These results extend earlier work of Erdős, Faudree, Rousseau and Schelp and of Balister, Schelp and Simonovits on so-called Ramsey size-linear graphs. Among others, we show that if $H$ is a subdivision of $K_4$ with at least $6$ vertices, then $R(H,F) = O(v(F) + e(F))$ for every graph $F$. We also conjecture that if $H$ is a connected graph with $e(H) - v(H) \leq \binom{k+1}{2} - 2$, then $R(H,K_n) = O(n^k)$. The case $k=2$ was proved by Erdős, Faudree, Rousseau and Schelp. We prove the case $k=3$.

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arXiv2023

Minimum Degree Threshold for $H$-factors with High Discrepancy

Given a graph $H$, a perfect $H$-factor in a graph $G$ is a collection of vertex-disjoint copies of $H$ spanning $G$. Kühn and Osthus showed that the minimum degree threshold for a graph $G$ to contain a perfect $H$-factor is either given by $1-1/χ(H)$ or by $1-1/χ_{cr}(H)$ depending on certain natural divisibility considerations. Given a graph $G$ of order $n$, a $2$-edge-coloring of $G$ and a subgraph $G'$ of $G$, we say that $G'$ has high discrepancy if it contains significantly (linear in $n$) more edges of one color than the other. Balogh, Csaba, Pluhár and Treglown asked for the minimum degree threshold guaranteeing that every 2-edge-coloring of $G$ has an $H$-factor with high discrepancy and they settled the case where $H$ is a clique. Here we completely resolve this question by determining the minimum degree threshold for high discrepancy of $H$-factors for every graph $H$.

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