arXiv subjects

Aleksa Džuklevski

Publications and source records attributed to Aleksa Džuklevski.

6 recordsLinked to original sources

arXiv2026

Many holes but no large one: maximizing $k$-holes while forbidding $(k+1)$-holes

We study the maximal number $m_{k,\ell,n}$ of empty convex $k$-gons ($k$-holes) determined by an $n$-point set in the plane in general position that contains no empty convex $\ell~$-gon, focusing on the first nontrivial case $\ell=k+1$. Our main result determines the exact value in the small-excess regime: for $n=k+a$ with $a\le k/2-1$, we prove $m_{k,k+1,k+a}=2^a.$ We also describe the extremal configurations attaining equality. Beyond this exact range, we provide upper and lower bounds in the proportional regime $n=αk$ and in the regime where $k$ is fixed and $n$ goes to infinity. In the last mentioned regime we prove that $m_{k,k+1,n}=Ω_k(n^{\lfloor\frac{k}{3}\rfloor})$ and $m_{k,k+1,n}=O_k(n^{\lfloor k/2\rfloor+1})$.

math.CO↗

arXiv2026

Erdős-Szekeres Maker-Breaker Games

We present new results on Maker-Breaker games arising from the Erdős-Szekeres problem in planar geometry. This classical problem asks how large a set in general position has to be to ensure the existence of $n$ points that are the vertices of a convex $n$-gon. Moreover, Erdős further extended this problem by asking what happens if we also require that this $n$-gon has an empty interior. In a 2-player Maker-Breaker setting, this problem inspires two main games. In both games, Maker tries to obtain an empty convex $k$-gon, while Breaker tries to prevent her from doing so. The games differ only in which points can comprise the winning $k$-gons: in the monochromatic version the points of both players can make up a $k$-gon, while in the bichromatic version only Maker's points contribute to such a polygon. Both settings are studied in this paper. We show that in the monochromatic game, Maker always wins. Even in a biased game where Breaker is allowed to place $s$ points per round, for any constant $s \geq 1$, Maker has a winning strategy. In the bichromatic setting, Maker still wins whenever Breaker is allowed to place $s$ points per round for any constant $s<2$. This settles an open problem posed by Aichholzer et al. (2019). Furthermore, we show that there are games that are not a lost cause for Breaker. Whenever $k\ge 8$ and Breaker is allowed to play 12 or more points per round, she has a winning strategy. We also consider the one-round bichromatic game (a.k.a.\ the offline version). In this setting, we show that Breaker wins if she can place twice as many points as Maker but if the bias is less than $2$, then Maker wins for large enough set of points.

math.CO↗

arXiv2025

Edge-Constrained Hamiltonian Paths on a Point Set

Let S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on S is a crossing-free geometric path such that every point of S is a vertex of the path. It is known that, if S is sufficiently large, there exist three edge-disjoint plane Hamiltonian paths on S. In this paper we study an edge-constrained version of the problem of finding Hamiltonian paths on a point set. We first consider the problem of finding a single plane Hamiltonian path pi with endpoints s, t in S and constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) ab in pi; (ii) ab not in pi. We characterize those quintuples (S, a, b, s, t) for which pi exists. Secondly, we consider the problem of finding two plane Hamiltonian paths pi_1, pi_2 on a set S with constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) pi_1 and pi_2 share no edges and ab is an edge of pi_1; (ii) pi_1 and pi_2 share no edges and none of them includes ab as an edge; (iii) both pi_1 and pi_2 include ab as an edge and share no other edges. In all cases, we characterize those triples (S, a, b) for which pi_1 and pi_2 exist.

cs.CG↗

arXiv2025

A $σ$-morphic convex protoset

We say that a tile is $σ$-morphic if it tiles the plane in exactly $\aleph_0$ many noncongruent ways (up to an isometry). It is an unsolved problem of whether a $σ$-morphic tile exist in the plane. In this note we present a construction of a set of convex tiles that is $σ$-morphic. The result is interesting since all the constructions of $σ$-morphic sets of tiles that arise in the literature make use of bumps and nicks, which necessarily make the tiles non-convex. We construct our set by cleverly dividing the tiles of the set of tiles discovered by Schmitt into convex tiles so that they behave in the same manner.

math.CO↗

arXiv2025

Classes of finite relational structures over finite languages have dual Ramsey degrees

Classical Ramsey theory has successfully extended to relational structures, yielding a wealth of results that have profoundly influenced other areas of mathematics. Interestingly, the same development has not occurred in the case of dual Ramsey theory. The main goal of this paper is to advance the dual Ramsey theory for finite relational structures with respect to natural structure-preserving maps. Tools from category theory prove instrumental in this endeavor, as was previously the case for finite algebraic systems where the dual Ramsey property had been established for every class of finite algebras coming from an equationally defined class. One cannot help but feel that dual Ramsey phenomena are deeply connected to categorical strategies.

math.CO↗

arXiv2025

Seeing is not believing in limited visibility cops and robbers

We consider the model of limited visibility Cops and Robbers, where the cops can only see within their $l$-neighbourhood. We prove that the number of cops needed to see the robber can be arbitrarily smaller than the number needed to capture the robber, answering an open question from the literature. We then consider how close we can get to seeing the robber when we do not have enough cops, along with a probabilistic interpretation.

math.CO↗