arXiv subjects

Aleksa Milojević

Publications and source records attributed to Aleksa Milojević.

22 records · Page 2Linked to original sources

arXiv2024

Canonical Ramsey numbers of sparse graphs

The canonical Ramsey theorem of Erdős and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy of $H$. The least such $N$ is called the Erdős-Rado number of $H$, denoted by $ER(H)$. Erdős-Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erdős-Rado numbers of sparse graphs. For example, we prove that if $H$ has bounded degree, then $ER(H)$ is polynomial in $|V(H)|$ if $H$ is bipartite, but exponential in general. We also study the closely-related problem of constrained Ramsey numbers. For a given tree $S$ and given path $P_t$, we study the minimum $N$ such that every edge-coloring of $K_N$ contains a monochromatic copy of $S$ or a rainbow copy of $P_t$. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse-Ackermann type.

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arXiv2024

Incidence bounds via extremal graph theory

The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as additive combinatorics and theoretical computer science. Assuming a standard non-degeneracy condition, i.e., that no $s$ points are contained in the intersection of $s$ hyperplanes, the currently best known upper bound on the number of incidences of $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ is $$O_{d, s}((mn)^{1-1/(d+1)}+m+n).$$ This bound by Apfelbaum and Sharir is based on geometrical space partitioning techniques, which apply only over the real numbers. In this paper, we propose a novel combinatorial approach to study such incidence problems over arbitrary fields. Perhaps surprisingly, this approach matches the best known bounds for point-hyperplane incidences in $\mathbb{R}^d$ for many interesting values of $m, n, d$, e.g. when $m=n$ and $d$ is odd. Moreover, in finite fields our bounds are sharp as a function of $m$ and $n$ in every dimension. We also study the size of the largest complete bipartite graph in point-hyperplane incidence graphs with a given number of edges and obtain optimal bounds as well. Additionally, we study point-variety incidences and unit-distance problem in finite fields, and give tight bounds for both problems under a similar non-degeneracy assumption. We also resolve Zarankiewicz type problems for algebraic graphs. Our proofs use tools such as induced Turán problems, VC-dimension theory, evasive sets and Hilbert polynomials. Also, we extend the celebrated result of Rónyai, Babai and Ganapathy on the number of zero-patterns of polynomials to the context of varieties, which might be of independent interest.

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arXiv2024

Point-variety incidences, unit distances and Zarankiewicz's problem for algebraic graphs

In this paper we study the number of incidences between $m$ points and $n$ varieties in $\mathbb{F}^d$, where $\mathbb{F}$ is an arbitrary field, assuming the incidence graph contains no copy of $K_{s,s}$. We also consider the analogous problem for algebraically defined graphs and unit distance graphs. First, we prove that if $\mathcal{P}$ is a set of $m$ points and $\mathcal{V}$ is a set of $n$ varieties in $\mathbb{F}^{D}$, each of dimension $d$ and degree at most $Δ$, and in addition the incidence graph is $K_{s,s}$-free, then the number of incidences satisfies $I(\mathcal{P}, \mathcal{V})\leq O_{d,Δ, s}(m^{\frac{d}{d+1}} n+m)$. This bound is tight when $s,Δ$ are sufficiently large with respect to $d$, with an appropriate choice of $\mathbb{F}=\mathbb{F}(m,n)$. We give two proofs of this upper bound, one based on the framework of the induced Turán problems and the other based on VC-dimension theory. In the second proof, we extend the celebrated result of Rónyai, Babai and Ganapathy on the number of zero-patterns of polynomials to the context of varieties, which might be of independent interest. We also resolve the problem of finding the maximum number of unit distances which can be spanned by a set of $n$ points $\mathcal{P}$ in $\mathbb{F^d}$ whose unit-distance graph is $K_{s, s}$-free, showing that it is $Θ_{d,s}(n^{2-\frac{1}{\lceil d/2\rceil +1}})$. Finally, we obtain tight bounds on the maximum number of edges of a $K_{s, s}$-free algebraic graph defined over a finite field, thus resolving the Zarankiewicz problem for this class of graphs.

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arXiv2022

Structured Codes of Graphs

We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.

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