arXiv subjects

Aleksa Milojević

Publications and source records attributed to Aleksa Milojević.

At least 19 recordsLinked to original sources

arXiv2026

Perfect matchings in hypergraphs and Feige's inequality

How large of a minimum degree does an $n$-vertex graph need before we are sure that it contains a perfect matching? Dirac's theorem states that a graph on an even number of vertices in which each vertex has degree at least $n/2$ has this property. In this short expository note, intended to be used in the classroom, we discuss how this statement generalizes to hypergraphs. In particular, we highlight an elegant connection between fractional perfect matchings in hypergraphs and a probabilistic inequality about nonnegative random variables, which was conjectured by Feige. We also present a very short self-contained proof of Feige's conjecture.

math.CO↗

arXiv2026

Sharp bounds for off-diagonal and tripartite canonical Ramsey numbers

The canonical Ramsey theorem establishes that for every positive integer $t$, there is a sufficiently large $n$ such that every edge-colouring of a complete graph $K_n$ contains a copy of $K_t$ that is canonically coloured, i.e. monochromatic, rainbow, or lexicographic. In this paper we investigate two variations of this theorem. First, we study canonical Ramsey numbers in the multipartite hypergraph setting. We prove that in every edge-colouring of the 3-uniform complete hypergraph on at least $t^{151t^2}$ vertices, there always exists a canonically coloured copy of $K_{t, t, t}^{(3)}$. This estimate is sharp up to the constant in the exponent. Second, we explore off-diagonal canonical Ramsey numbers. Let $ER(a, b, c)$ denote the minimum number of vertices $n$ required to guarantee a monochromatic $K_a$, a lexicographic $K_b$, or a rainbow $K_c$ in every edge-colouring of the complete graph $K_n$. We establish sharp bounds for these numbers across different parameter regimes. Specifically, we prove that $ER(a, b, c)\geq c^{Ω(ab)}$ when $c$ is sufficiently large relative to $a$, and that $ER(a, b, c)\leq 2^{O_b(a)}$ when $b\ge 4$ is a fixed constant and $c\le a$. Finally, we analyse the behaviour of the function when avoiding lexicographic triangles (i.e. $b=3$), showing that $ER(a, 3, c)\le (a-1)(c-2)+O(c^6)$.

math.CO↗

arXiv2026

Minimising the harmonic sum of cycle lengths

A central theme in extremal graph theory is to understand the relationship between the density of a graph and the richness of its cycle length spectrum, which is the set of distinct cycle lengths occurring in the graph. In 1966, Erdős and Hajnal suggested studying $s(G):=\sum_{\ell\in{C}(G)}1/\ell$ as a measure of the richness of the cycle length spectrum ${C}(G)$ of a graph $G$. Through a series of increasingly strong conjectures, Erdős suggested that the complete bipartite graphs minimise $s(G)$ among all graphs $G$ with the same average degree. The sharpest such conjecture, from 1981, states that the graph $K_{k, n-k}$ minimises $s(G)$ among all $n$-vertex graphs with at least $k(n-k)$ edges (where $k\leq n/2$). We prove this conjecture for all sufficiently large $k$, by showing the stronger statement that any $n$-vertex graph $G$ with $e(G)>(k-1)(n-k+1)$ and $n\geq 2k$ satisfies $s(G)\geq\sum_{\ell=2}^{k}1/(2\ell)$. Moreover, we show that the complete bipartite graph $K_{k,n-k}$ is the unique graph with at least $k(n-k)$ edges that achieves equality here.

math.CO↗

arXiv2026

Spanning subhypergraphs with degree constraints

An old result of Tutte states that any $d$-regular graph contains a spanning subgraph in which every vertex has degree $k$ or $k+1$, for every $1\leq k\leq d$. We generalize this statement to hypergraphs, showing, for example, that every $3$-uniform $d$-regular hypergraph contains a subgraph in which all degrees are $k, k+1$ or $k+2$, for every $1\leq k\leq d$. This statement is best possible in the sense that the corresponding statement with only two allowed consecutive values is not true. We provide generalizations of this statement to higher uniformities and discuss several open problems.

math.CO↗

arXiv2026

Distinguishability threshold for random geometric graphs

The spherical random geometric graph $G(n,d,p)$ is obtained by sampling $n$ independent points uniformly on the unit sphere $\mathbb{S}^{d-1}\subseteq\mathbb{R}^d$ and joining pairs of points which are sufficiently close, where the threshold is chosen so that the edge probability is $p$. The central question related to this model, and to a broad class of other models, is the following: when does the underlying geometry affect the resulting graph in a way which makes it distinguishable from the Erdős--Rényi random graph $G(n,p)$, as measured in total variation distance? The precise answer to this question was conjectured by Bubeck, Ding, Eldan, and Rácz, who predicted that $G(n,d,p)$ and $G(n,p)$ are indistinguishable precisely when $d \gg n^3p^3(\log p^{-1})^3$, and provided a test for distinguishing these models in the low-dimensional regime. Although this conjecture attracted considerable attention from researchers in probability, theoretical computer science, and high-dimensional statistics, it was previously fully proved only in the constant-density case. In this paper, we resolve the distinguishability conjecture in the broad range $1/3 \geq p \geq n^{-1/5} \text{polylog}(n)$. The key ingredient of our proof is a stronger statement which gives a precise asymptotic formula for the probability that $G(n,d,p)$ realizes a prescribed graph $H$: above the conjectured threshold, this probability is at most $(1+o(1))$ times the corresponding probability for $G(n,p)$, with the signed triangle count of $H$ appearing as the leading correction term.

math.PR↗

arXiv2026

Nearly tight bounds for induced subdivisions

Subdivisions of complete graphs play a central role in combinatorics, having deep connections to structural, extremal, and topological aspects of graph theory. A celebrated conjecture of Mader, proved independently by Bollobás and Thomason and by Komlós and Szemerédi, states that every graph of average degree of order $h^2$ contains a subdivision of $K_h$. In this paper, we consider the induced variant of this problem. A theorem of Kühn and Osthus implies that, for every fixed graph $H$ and every $s\ge 1$, graphs of sufficiently large average degree contain either a copy of $K_{s,s}$ or an induced subdivision of $H$. However, even for $H=K_h$, the best previous quantitative bounds were far from optimal. We prove nearly tight bounds for forcing induced subdivisions of $K_h$. We show that every $K_{s,t}$-free graph of average degree $Ω_{s,t}(h^{2(s-1)}\log^{7(s-1)} h)$ contains an induced subdivision of $K_h$, and that every $C_{2k}$-free graph with $k \geq 3$ and average degree $Ω_k(h\log^5 h)$ contains an induced subdivision of $K_h$. These bounds substantially improve the previously known results and are nearly optimal in both settings. They also hold if $K_h$ is replaced by any other graph on $h$ vertices.

math.CO↗

arXiv2026

Gaussian random graphs and Ramsey numbers

We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify their analysis significantly. As a consequence of this simpler analysis, we also obtain better quantitative bounds.

math.CO↗

arXiv2026

Communication Complexity of Disjointness under Product Distributions

Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the famous works of Kalyanasundaram-Schnitger and Razborov in 1992. However, the question of obtaining tight bounds for product distributions persisted until the more recent work of Bottesch, Gavinsky, and Klauck resolved it. In this note we revisit this classical problem and give a short, streamlined proof of the best bounds, with improved quantitative dependence on the error parameter. Our approach is based on a simple combinatorial lemma that may be of independent interest: if two sets drawn independently from two distributions are disjoint with non-negligible probability, then one can extract two subfamilies of reasonably large measure that are fully cross-disjoint (equivalently, a large monochromatic rectangle for disjointness).

cs.CC↗

arXiv2026

Universality for transversal Hamilton cycles in random graphs

A tuple $(G_1,\dots,G_n)$ of graphs on the same vertex set of size $n$ is said to be Hamilton-universal if for every map $χ: [n]\to[n]$ there exists a Hamilton cycle whose $i$-th edge comes from $G_{χ(i)}$. Bowtell, Morris, Pehova and Staden proved an analog of Dirac's theorem in this setting, namely that if $δ(G_i)\geq (1/2+o(1))n$ then $(G_1,\dots,G_n)$ is Hamilton-universal. Combining McDiarmid's coupling and a colorful version of the Friedman-Pippenger tree embedding technique, we establish a similar result in the setting of sparse random graphs, showing that there exists $C$ such that if the $G_i$ are independent random graphs sampled from $G(n,p)$, where $p\geq C\log n/n$, then $(G_1,\dots,G_n)$ is Hamilton-universal with high probability.

math.CO↗

arXiv2025

From small eigenvalues to large cuts, and Chowla's cosine problem

We prove that every graph with average degree $d$ and smallest adjacency eigenvalue $|λ_n|\leq d^γ$ contains a clique of size $d^{1-O(γ)}$. A simple corollary of this yields the first polynomial bound for Chowla's cosine problem (1965): for every finite set $A\subseteq \mathbb{Z}_{>0}$, the minimum of the cosine polynomial satisfies $$\min_{x\in [0, 2π]}\sum_{a\in A}\cos(ax)\leq -|A|^{1/10-o(1)}.$$ Another application makes significant progress on the problem of MaxCut in $H$-free graphs initiated by Erdős and Lovász in the 1970's. We show that every $m$-edge graph with no clique of size $m^{1/2-δ}$ has a cut of size at least $m/2+m^{1/2+\varepsilon}$ for some $\varepsilon=\varepsilon(δ)>0$.

math.CO↗

arXiv2025

Long induced paths in $K_{s, s}$-free graphs

More than 40 years ago, Galvin, Rival and Sands showed that every $K_{s, s}$-free graph containing an $n$-vertex path must contain an induced path of length $f(n)$, where $f(n)\to \infty$ as $n\to \infty$. Recently, it was shown by Duron, Esperet and Raymond that one can take $f(n)=(\log \log n)^{1/5-o(1)}$. In this note, we give a short self-contained proof that a $K_{s, s}$-free graphs with an $n$-vertex path contains an induced path of length at least $(\log \log n)^{1-o(1)}$. Combined with the recent remarkable example of Couëtoux, Defrain, and Raymond, which provides an upper bound of $O((\log \log n)^{1+o(1)})$, this essentially resolves this old problem.

math.CO↗

arXiv2025

Cyclic subsets of tournaments

Let $G$ be a Dirac graph, and let $S$ be a vertex subset of $G$, chosen uniformly at random. How likely is the induced subgraph $G[S]$ to be Hamiltonian? This question, proposed by Erdős and Faudree in 1996, was recently resolved by Draganić, Keevash and Müyesser, in the setting of graphs. In this paper, we study a similar question for tournaments -- if $T$ is a tournament of high minimum degree, how likely is it for a random induced subtournament of $T$ to be Hamiltonian? We prove an optimal bound on this probability, and extend the results to the regime where the subset is not sampled uniformly at random, but according to a $p$-biased measure.

math.CO↗

arXiv2025

Beyond the MaxCut problem in $H$-free graphs

In a recent breakthrough, Zhang proves that if $G$ is an $H$-free graph with $m$ edges, then $G$ has a cut of size at least $m/2+c_Hm^{0.5001}$, making a significant step towards a well known conjecture of Alon, Bollobás, Krivelevich and Sudakov. We show that the methods of Zhang can be further boosted, and prove the following strengthening. If $G$ is a graph with $m$ edges and no clique of size $m^{1/2-δ}$, then $G$ has a cut of size at least $m/2+m^{1/2+\varepsilon}$ for some $\varepsilon=\varepsilon(δ)>0$. In addition, we sharpen another result of Zhang by proving that if $G$ is an $n$-vertex $m$-edge graph with MaxCut of size at most $m/2+n^{1+\varepsilon}$ (or its smallest eigenvalue $λ_n$ satisfies $|λ_n|\leq n^{\varepsilon}$), then $G$ is $n^{-\varepsilon}$-close to the disjoint union of cliques for some absolute constant $\varepsilon>0$.

math.CO↗

arXiv2025

Power saving for the Brown-Erdős-Sós problem

Let $f(n, v, e)$ denote the maximum number of edges in a 3-uniform hypergraph on $n$ vertices which does not contain $v$ vertices spanning at least $e$ edges. A central problem in extremal combinatorics, famously posed by Brown, Erdős and Sós in 1973, asks whether $f(n, e+3, e)=o(n^2)$ for every $e \ge 3$. A classical result of Sárközy and Selkow states that $f(n, e+\lfloor \log_2 e\rfloor+2, e)=o(n^{2})$ for every $e \ge 3$. This bound was recently improved by Conlon, Gishboliner, Levanzov and Shapira. Motivated by applications to other problems, Gowers and Long made the striking conjecture that $f(n, e+4, e)=O(n^{2-\varepsilon})$ for some $\varepsilon=\varepsilon(e)>0$. Conlon, Gishboliner, Levanzov and Shapira, and later, Shapira and Tyomkyn reiterated the following approximate version of this problem. What is the smallest $d(e)$ for which $f(n, e+d(e), e)=O(n^{2-\varepsilon})$ for some $\varepsilon=\varepsilon(e)>0$? In this paper, we prove that for each $e\geq 3$ we have $f(n, e+\lfloor \log_2 e\rfloor +38, e)=O(n^{2-\varepsilon})$ for some $\varepsilon>0$. This shows that one can already obtain power saving near the Sárközy-Selkow bound at the cost of a small additive constant.

math.CO↗

arXiv2025

$C_4$-free subgraphs of high degree with geometric applications

The Zarankiewicz problem, a cornerstone problem in extremal graph theory, asks for the maximum number of edges in an $n$-vertex graph that does not contain the complete bipartite graph $K_{s,s}$. While the problem remains widely open in the case of general graphs, the past two decades have seen significant progress on this problem for various restricted graph classes -- particularly those arising from geometric settings -- leading to a deeper understanding of their structure. In this paper, we develop a new structural tool for addressing Zarankiewicz-type problems. More specifically, we show that for any positive integer $k$, every graph with average degree $d$ either contains an induced $C_4$-free subgraph with average degree at least $k$, or it contains a $d$-vertex subgraph with $Ω_k(d^2)$ edges. As an application of this dichotomy, we propose a unified approach to a large number of Zarankiewicz-type problems in geometry, obtaining optimal bounds in each case.

math.CO↗

arXiv2025

Kővári-Sós-Turán theorem for hereditary families

The celebrated Kővári-Sós-Turán theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose the systematic study of this phenomenon, and state the conjecture that if $H$ is a bipartite graph, then an induced $H$-free and $K_{s,s}$-free graph cannot have much more edges than an $H$-free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by $k$ on one side, obtaining in all the cases similar bounds as in the non-induced setting. Our results also have applications to the Erdős-Hajnal conjecture, the problem of finding induced $C_4$-free subgraphs with large degree and bounding the average degree of $K_{s, s}$-free graphs which do not contain induced subdivisions of a fixed graph.

math.CO↗

arXiv2024

Larger Nearly Orthogonal Sets over Finite Fields

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove that for every prime $p$ there exists some $δ= δ(p)>0$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k$, there exists a $k$-nearly orthogonal set of at least $d^{δ\cdot k/\log k}$ vectors of $\mathbb{F}^d$. The size of the set is optimal up to the $\log k$ term in the exponent. We further prove two extensions of this result. In the first, we provide a large set ${\cal A}$ of non-self-orthogonal vectors of $\mathbb{F}^d$ such that for every two subsets of ${\cal A}$ of size $k+1$ each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every $k+1$ vectors of the produced set ${\cal A}$ include $\ell+1$ pairwise orthogonal vectors for an arbitrary fixed integer $1 \leq \ell \leq k$. The proofs involve probabilistic and spectral arguments and the hypergraph container method.

math.CO↗

arXiv2024

Disjoint pairs in set systems and combinatorics of low rank matrices

We study and solve several problems in two closely related settings: set families in $2^{[n]}$ with many disjoint pairs of sets and low rank matrices with many zero entries. - More than 40 years ago, Daykin and Erdős asked for the maximum number of disjoint pairs of sets in a family $F\subseteq 2^{[n]}$ of size $2^{(1/2+δ)n}$ and conjectured it contains at most $o(|F|^2)$ such pairs. This was proven by Alon and Frankl in 1985. In this paper we completely resolve this problem, proving an optimal dependence of the number of disjoint pairs on the size of family $F$. We also prove the natural variant of the Daykin-Erdős conjecture in which disjoint pairs are replaced by pairs with intersection $λ\neq 0$. - Motivated by a conjecture of Lovett related to the famous log-rank conjecture, Singer and Sudan asked to show that for two families $A, B \subseteq 2^{[n]}$ with a positive constant fraction of set pairs $(a,b)\in A\times B$ being disjoint, there are $R\subset A$ and $S\subset B$ such that all set pairs $(r, s)\in R\times S$ are disjoint, and $|R|\geq 2^{-O(\sqrt{n})}|A|$ and $|S|\geq 2^{-O(\sqrt{n})}|B|$. We prove this conjecture in a strong quantitative form. - We prove the following generalizations of the best known bounds for the log-rank conjecture. If $M$ is an $n\times n$ non-negative integer matrix of rank $r$ in which the average of the entries is $\varepsilon\leq 1/2$, then $M$ contains an all-zero submatrix of size at least $2^{-O(\sqrt{\varepsilon r})}n$. Unlike the known bounds for the log-rank conjecture, this result is optimal. Moreover, using similar methods, we also prove that any $n\times n$ matrix of rank $r$ with entries from $\{0,\dots,t\}$ contains a constant submatrix of size at least $2^{-O(t\sqrt{r})}n$. Our proofs use probabilistic, entropy and discrepancy methods and explore connections to additive combinatorics and coding theory.

math.CO↗