arXiv subjects

Balázs Patkós

Publications and source records attributed to Balázs Patkós.

At least 19 recordsLinked to original sources

arXiv2026

Forbidden subposet problems in the linear lattice

We study weak and strong forbidden subposet problems in the linear lattice $L_n(q)$. We show that the $q$-analogues of the Bukh--Griggs--Lu conjecture fail for every prime power $q$. If $q\ge 3$ and $n$ and $d$ have opposite parity, then $La_q(n,L_d(q))=La_q^*(n,L_d(q))=Σ_q(n,d)$. We also show that $La_q(n,D_s)=La_q^*(n,D_s)=Σ_q(n,2)$ for $2\le s\le q$ and all $n$, and for $s=q+1$ when $n$ is odd. For $n$ even, a maximum strong $D_{q+1}$-free family can be contained from the three middle layers. The case $s=2$ settles the $q$-analog of the diamond conjecture in the affirmative.

math.CO↗

arXiv2026

Embedding rooted blow-ups of tree posets

A tree poset $T$ is a poset whose Hasse diagram is a tree. Bukh proved that if a family $\mathcal F\subseteq 2^{[n]}$ contains $(h(T)-1+\varepsilon)\binom{n}{\lfloor \frac{n}{2}\rfloor}$ sets, then $\mathcal F$ contains a weak copy of $T$, where $h(T)$ is the height of $T$, the number of elements in a longest chain of $T$. Several strengthenings and generalizations of this result have been obtained. We prove the following robust variant. For a tree poset $T$ and $x\in T$, the $b$-blow-up $T(x,b)$ rooted at $x$ is the tree poset that we obtain from $T$ by replacing every element $u$ by $b^d$ new elements, where $d$ is the distance $d(x,u)$ in the Hasse diagram of $T$ and an edge $uv$ with $v$ being closer to $x$ is replaced by edges such that every new copy of $v$ is joined to $b$ new copies of $u$ such that these new copies form pairwise disjoint sets for the copies of $v$. We prove that for any tree poset $T$, $x\in T$, and $\varepsilon>0$ there exists $δ$ such that if $\mathcal F\subseteq 2^{[n]}$ contains $(h(T)-1+\varepsilon)\binom{n}{\lfloor \frac{n}{2}\rfloor}$ sets, then $\mathcal F$ contains a weak copy of $T(x,\lfloor δn\rfloor)$. This settles a conjecture of Treglown and the author. As applications, we derive the known asymptotic counting and random versions of Bukh's theorem from this stronger embedding result, and obtain new maximal anti-Ramsey results for tree posets.

math.CO↗

arXiv2026

Maximal anti-Ramsey problems for posets

We study the forbidden poset analog of the maximal anti-Ramsey problem introduced for graphs by Burr, Erd\H os, Graham, and Sós. For integers $m\le 2^n$ and poset $P=(P,\preceq)$, we introduce $\mathrm{ar_m}(n,m,P)$ (and $\mathrm{ar^*_m}(n,m,P)$) to denote the minimum integer $k$ such that there exists a family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|=m$ and a coloring $\mathcal{F}\rightarrow [k]$ with all weak (strong) copies of $P$ being rainbow. As long as there exist $P$-free families of size $m$, these parameters equal 1. It is known that the largest size $La(n,P)$ ($La^*(n,P)$) of weak (strong) $P$-free families has order of magnitude $Θ(\binom{n}{\lfloor n/2\rfloor})$ unless $P$ is the antichain $A_k$ on $k$ elements. In this paper we study $\mathrm{ar_m}(n,m,P)$ and $\mathrm{ar^*_m}(n,m,P)$ in two regimes of $m$. We determine the asymptotics of these parameters for all posets $P$ when $m=2^n$. We also consider the case $m=Θ(\binom{n}{\lfloor n/2\rfloor})$. It is shown that for any connected poset $P$ and integer $k$, there exist integers $m_{P,k}$ and $m^*_{P,k}$ such that to color the middle $k$ layers of the Boolean lattice with all weak or strong copies of $P$ being rainbow, one needs $Θ(n^{m_{P,k}})$ or $Θ(n^{m^*_{P,k}})$ colors. For tree posets $T$, one has $m_{T,k}=m^*_{T,k}$. We conjecture that for any tree poset $T$, and positive real $\varepsilon$, $\mathrm{ar_m}(n,m,T),\mathrm{ar^*_m}(n,m,T)=Ω(n^{m_{T,k}})$ holds provided $m\ge (k-1+\varepsilon)\binom{n}{\lfloor n/2\rfloor}$. We prove our conjecture on $\mathrm{ar_m}(n,m,T)$ for an infinite class of tree posets.

math.CO↗

arXiv2026

Crown-free families and forbidden subposets with $e(P)\in \{1,2\}$

The maximum size of a weak $P$-free family $\mathcal{F}\subseteq 2^{[n]}$ is denoted by $La(n,P)$. Let $e(P)$ denote the maximum integer $k$ such that the union of any $k$ consecutive layers of $2^{[n]}$ is weak $P$-free. In recent years, multiple examples of posets with $e(P)<π^-(P):=\liminf_{n\to\infty} \frac{La(n,P)}{\binom{n}{\lfloor \frac{n}{2}\rfloor}}$ have been found. We add several further posets with $e(P)=1$ to this list. We define a family $\mathcal{F}\subseteq 2^{[n]}$ of size at least $(1.22+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ that is weak $O_6$-free, where $O_6$ is the six-element crown poset. We also show an infinite set of posets $P$ with $1=e(P)<π^-(P)$ that are minimal with respect to this property. Finally, we consider how far apart $e(P)$ and $π^-(P)$ can be. We prove that for every fixed finite poset $P$ with $e(P)=1$, there is a constant $δ_P>0$ such that $La(n,P)\le(2-δ_P+o(1))\binom{n}{\lfloor n/2\rfloor}$. The value 2 is optimal: explicit vertex-edge incidence posets with $e(P)=1$ have $π^-(P)$ values tending to $2$. In contrast, for every $K>0$ we construct a finite poset $P$ with $e(P)=2$ and lower density greater than $K$.

math.CO↗

arXiv2026

On the damage number of graphs

We study a variant of Cops and Robbers in which the robber attempts to visit as many vertices of the graph as possible without being captured, while the cop aims to keep the robber confined to a small set of vertices. The \textit{damage number} of a graph $G$, introduced by Cox and Sanaei in 2019, is the maximum number of vertices the robber can visit in a game of Cops and Robbers on $G$. In this paper, we determine damage numbers for several classes of graphs, including hypercubes, Hamming graphs, Johnson graphs, incidence graphs of projective planes, and Erdős-Rényi random graph $G(n,p)$, for $p \gg \log^{2/5}(n) / n^{1/5}$. We also show that the problem of determining the damage number of a graph is {\sf PSPACE}-complete.

math.CO↗

arXiv2026

On the maximum size of $B_3$-free and $D_s$-free families

For a poset $P$, let $e(P)$ ($e^*(P)$) denote largest positive integer $k$ such that the union of the $k$ middle layers of $2^{[n]}$ does not contain a weak (strong) copy of $P$. Ellis, Ivan, and Leader showed the existence of posets $P$ for which there exists a positive real $\varepsilon_P$ such that $La(n,P)\ge (e(P)+\varepsilon_P)\binom{n}{\lfloor n/2\rfloor}$ and $La^*(n,P)\ge (e^*(P)+\varepsilon_P)\binom{n}{\lfloor n/2}$ hold, where $La(n,P)$ ($La^*(n,P)$) denotes the maximum size of a weak (strong) $P$-free family $\mathcal{F}\subseteq 2^{[n]}$. More precisely, they showed that $P=B_d$ are such posets for all $d\ge 4$, where $B_d$ is the Boolean lattice ordered by inclusion. Tompkins showed that the diamond $B_2$ is also such a poset. We apply his method to settle the case of the last Boolean poset $B_3$. We show that there exists a positive $\varepsilon$ such that $$La^*(n,B_3)\ge La(n,B_3)\ge La(n,D_6)\ge (3+\varepsilon)\binom{n}{\lfloor n/2\rfloor},$$ where $D_s$ is the poset on $s+2$ elements $a<b_1,\dots,b_s<c$. Consider the intervals $I_m=[2^{m-1}-1,2^m-2]$, $I^*_m=[\binom{m-1}{\lfloor \frac{m-1}{2}\rfloor}+1,\binom{m}{\lfloor \frac{m}{2}\rfloor}]$. It is known that for values $s$ in the major initial parts of $I_m$ and $I_m^*$, one has $La(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ and $La^*(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$. The above equalities do not hold for the largest elements of the intervals, thus there exist $s_m\in I_m, s^*_m\in I^*_m$ such that for $s\in I_m$ we have $La(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ if and only if $s<s_m$ and for $s\in I^*_m$ we have $La^*(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ if and only if $s<s^*_m$. Modifying previous constructions, we obtain upper bounds on $s_m$ and $s^*_m$.

math.CO↗

arXiv2026

Turán problems for simplicial complexes

An abstract simplicial complex $\mathbf{F}$ is a non-uniform hypergraph without isolated vertices, whose edge set is closed under taking subsets. The extremal number $\mathrm{ex}(n,\mathbf{F})$ is defined as the maximum number of edges in an $n$-vertex $\mathbf{F}$-free simplicial complex. Although Turán-type problems for simplicial complexes have long appeared in extremal set theory, a systematic study of $\mathrm{ex}(n,\mathbf{F})$ was initiated only recently by Conlon, Piga, and Schülke. In contrast to uniform hypergraphs, even the order of magnitude of $\mathrm{ex}(n,\mathbf{F})$ remains unknown for most simplicial complexes. In this paper, we present a general framework for estimating $\mathrm{ex}(n,\mathbf{F})$ via generalised Turán numbers of associated hypergraphs. Using this approach, we determine the asymptotic behaviour, and in some cases the exact value, of $\mathrm{ex}(n,\mathbf{F})$ for broad classes of simplicial complexes, extending a result of Conlon, Piga, and Schülke. We also exhibit simplicial complexes whose extremal numbers are not governed by the corresponding generalised Turán numbers, and determine their extremal numbers asymptotically. In addition, we study how the extremal number changes when a new edge is added to the forbidden simplicial complex, and obtain a tight bound for this behaviour.

math.CO↗

arXiv2026

Rainbow Turán problems for forbidden subposets

A family $\mathcal{G}$ of sets is a copy of a poset $(P,\leqslant)$ if $(\mathcal{G},\subseteq)$ is isomorphic to $(P,\leqslant)$. The forbidden subposet problem asks for determining $La^*(n,P)$, the maximum size of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain any copy of $P$. We study the rainbow version of this problem: what is the maximum size $La_R^*(n,P)$ of a family $\mathcal{F}=\cup_{i=1}^mA^i$ such that all $A^i$ are antichains and there is no copy of $P$ with all sets coming from distinct $A^i$ or equivalently $\mathcal{F}$ admits a proper coloring (sets $F\subset F'$ must receive different colors) with no rainbow copy of $P$. A poset $(Q,\leqslant')$ rainbow forces $(P,\leqslant)$ if any proper coloring $c$ of $Q$ ($q\leqslant' q'$ or $q'\leqslant' q$ implies $c(q)\neq c(q')$) admits a rainbow copy of $P$. We establish connection between the $La^*$ and the $La^*_R$ functions via poset rainbow forcing, determine the asymptotics of $La_R^*(n,T)$ for all tree posets and obtain further exact or asymptotic results for antichains and complete bipartite posets.

math.CO↗

arXiv2026

Edge mappings of graphs: Ramsey type parameters

In this paper, we launch a systematic study of parameters concerning edge mappings of graphs. Inspired by Ramsey's theorem, the quantity $m(G,H)$ is defined to be the smallest integer $n$ such that for every $f:E(K_n)\rightarrow E(K_n)$ either there is an $f$-fixed copy of $G$ with $f(e)=e$ for all $e\in E(G)$, or an $f$-free copy of $H$ with $f(e)\notin E(H)$ for all $e\in E(H)$. Incorporating new ideas, we extend many old results from the 1980s and prove many new exact results, mostly concerning $m(T,K_r)$, where $T$ is a tree, and $m(G,K_{1,r})$. We also study further related parameters, most of them introduced in the 1980s, and obtain substantial progress regarding these parameters.

math.CO↗

arXiv2026

Anti-Ramsey forbidden poset problems

A family $\mathcal{G}$ of sets is a weak copy of a poset $P$ if there is a bijection $f:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ implies $f(p)\subseteq f(q)$. If $f$ satisfies $p\leqslant q$ if and only if $f(p)\subseteq f(q)$, the $\mathcal{G}$ is a strong copy of $P$. We study the anti-Ramsey numbers $\mathrm{ar}(n,P), \mathrm{ar^*}(n,P)$, the maximum number of colors used in a coloring of $2^{[n]}$ that does not admit a rainbow weak or strong copy of $P$, respectively. We establish connections to the well-studied extremal numbers $\mathrm{La}(n,P)$ and $\mathrm{La^*}(n,P)$ and determine asymptotically $\mathrm{ar^*}(n,T)$ for all tree posets $T$ and $\mathrm{ar^*}(n,O_{2k})$ for all crown posets $O_{2k}$.

math.CO↗

arXiv2026

Chain Covers in the Boolean Lattice

For integers $1\le r\le n+1$, let $N(n,r)$ denote the least number of chains in the Boolean lattice $B_n=2^{[n]}$ that cover every strict $r$-term chain. The case $r=1$ is the classical chain-decomposition problem and is generalizing Dilworth's theorem and Sperner's theorem. We study two complementary regimes. First, when $r>1$ is fixed and $n\to\infty$. Let $$M(n,r):= \max_{\substack{ a_0+\cdots+a_r=n a_0,a_r\ge 0,\ a_i\ge 1\ (1\le i\le r-1) }} \binom{n}{a_0,\ldots,a_r}.$$ We prove that lower and upper bounds which differ only by a logarithmic factor: \[ M(n,r)\le N(n,r)\le \left(\frac r2+o(1)\right)\log n\cdot M(n,r). \] Second, we consider the near-maximal regime $N(n,n-t)$, where $t>0$ is fixed. We prove that \[ N(n,n-t)= (γ_{t-1}+o(1))n!, \] where $γ_d$ is the limit of the density of a minimum size subset of the hypercube $Q_n$ that meets all $d$-dimensional subcube.

math.CO↗

arXiv2025

Poset saturation of unions of chains

A family $\mathcal{G}$ of sets is a(n induced) copy of a poset $P=(P,\leqslant)$ if there exists a bijection $b:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ holds if and only if $b(p)\subseteq b(q)$. The induced saturation number sat$^*(n,P)$ is the minimum size of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain any copy of $P$, but for any $G\in 2^{[n]}\setminus \mathcal{F}$, the family $\mathcal{F}\cup \{G\}$ contains a copy of $P$. We consider sat$^*(n,P)$ for posets $P$ that are formed by pairwise incomparable chains, i.e. $P=\bigoplus_{j=1}^mC_{i_j}$. We make the following two conjectures: (i) sat$^*(n,P)=O(n)$ for all such posets and (ii) sat$^*(n,P)=O(1)$ if not all chains are of the same size. (The second conjecture is known to hold if there is a unique longest among the chains.) We verify these conjectures in some special cases: we prove (i) if all chains are of the same length, we prove (ii) in the first unknown general case: for posets $2C_k+C_1$. Finally, we give an infinite number of examples showing that (ii) is not a necessary condition for sat$^*(n,P)=O(1)$ among posets $P=\bigoplus_{j=1}^mC_{i_j}$: we prove sat$^*(n,(\binom{2t}{t}+1)C_2)=O(1)$ for all $t\ge 1$.

math.CO↗

arXiv2025

Finding the diameter of a tree with distance queries

We study the number of distance queries needed to identify certain properties of a hidden tree $T$ on $n$ vertices. A distance query consists of two vertices $x,y$, and the answer is the distance of $x$ and $y$ in $T$. We determine the number of queries an optimal adaptive algorithm needs to find two vertices of maximal distance up to an additive constant, and the number of queries needed to identify the hidden tree asymptotically. We also study the non-adaptive versions of these problems, determining the number of queries needed exactly.

cs.DS↗

arXiv2025

Generalized saturation game

We study the following game version of the generalized graph Turán problem. For two fixed graphs F and H, two players, Max and Mini, alternately claim unclaimed edges of the complete graph Kn such that the graph G of the claimed edges must remain F-free throughout the game. The game ends when no further edges can be claimed, i.e. when G becomes F-saturated. The H-score of the game is the number of copies of H in G. Max aims to maximize the H-score, while Mini wants to minimize it. The H-score of the game when both players play optimally is denoted by s1(n, #H, F) when Max starts, and by s2(n, #H, F) when Mini starts. We study these values for several natural choices of F and H.

math.CO↗

arXiv2025

Identification of a monotone Boolean function with $k$ "reasons" as a combinatorial search problem

We study the number of queries needed to identify a monotone Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$. A query consists of a 0-1-sequence, and the answer is the value of $f$ on that sequence. It is well-known that the number of queries needed is $\binom{n}{\lfloor n/2\rfloor}+\binom{n}{\lfloor n/2\rfloor+1}$ in general. Here we study a variant where $f$ has $k$ ``reasons'' to be 1, i.e., its disjunctive normal form has $k$ conjunctions if the redundant conjunctions are deleted. This problem is equivalent to identifying an upfamily in $2^{[n]}$ that has exactly $k$ minimal members. We find the asymptotics on the number of queries needed for fixed $k$. We also study the non-adaptive version of the problem, where the queries are asked at the same time, and determine the exact number of queries for most values of $k$ and $n$.

math.CO↗

arXiv2025

Bipartite Turán number of trees

We start a systematic investigation concerning bipartite Turán number for trees. For a graph $F$ and integers $1 \leq a \leq b$ we define: $(i)$\quad $ex_b(a, b, F)$ is the largest number of edges that an $F$-free bipartite graph can have with part sizes $a$ and $b$. We write $ex_b(n, F)$ for $ex_b(n, n, F)$. $(ii)$\quad $ex_{b,c}(a, b, F)$ is the largest number of edges that an $F$-free connected, bipartite graph can have with part sizes $a$ and $b$. We write $ex_{b,c}(n, F)$ for $ex{b,c}(n, n, F)$. Both definitions are similar for a family $\mathcal{F}$ of graphs. We prove general lower bounds depending on the maximum degree of $F$, as well as on the cardinalities of the two vertex classes of $F$. We derive upper and lower bounds for $ex_b(n,F)$ in terms of $ex(2n,F)$ and $ex(n, F)$, the corresponding classical (not bipartite) Turán numbers. We solve both problems for various classes of graphs, including all trees up to six vertices for any $n$, for double stars $D_{s ,t}$ if $a \geq f(s,t )$, for some families of spiders, and more. We use these results to supply an answer to a problem raised by L. T. Yuan and X. D. Zhang [{\it Graphs and Combinatorics}, 2017] concerning $ex_b( n, \mathcal{T}_{k,\ell} )$, where $\mathcal{T}_{k,\ell}$ is the family of all trees with vertex classes of respective cardinalities $k$ and $\ell$. The asymptotic worst-case ratios between Turán-type functions are also inverstigated.

math.CO↗

arXiv2025

Size, diversity, minimum degree, sturdiness, dömdödöm

For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-dömdödöm} of a family $\mathcal{F}\subseteq 2^{[n]}$ is $β_{p,q}(\mathcal{F})=\min\{|\mathcal{F}(A,\overline{B})|:|A|=p,|B|=q, A\cap B=\emptyset, A,B\subseteq [n]\} $. This definition encompasses size, diversity, minimum degree, and sturdiness as special cases. We investigate the maximum possible value $β_{p,q}(n,k)$ of $β_{p,q}(\mathcal{F})$ over all $k$-uniform intersecting families $\mathcal{F}\subset 2^{[n]}$. We determine the order of magnitude of $β_{p,q}(n,k)$ for all fixed $p,q,k$. We relate the asymptotics of $β_{p,q}(n,k)$ to the constant value of $β_{0,q}(n,q+1)$ and establish $β_{p,1}(n,k)=\binom{n-3-p}{k-2-p}$ and $β_{p,2}(n,k)=2\binom{n-5}{k-3-p}-\binom{n-7}{k-5-p}$ if $n$ is large enough.

math.CO↗

arXiv2024

A note on the Erdős Matching Conjecture

The Erd\H os Matching Conjecture states that the maximum size $f(n,k,s)$ of a family $\mathcal{F}\subseteq \binom{[n]}{k}$ that does not contain $s$ pairwise disjoint sets is $\max\{|\mathcal{A}_{k,s}|,|\mathcal{B}_{n,k,s}|\}$, where $\mathcal{A}_{k,s}=\binom{[sk-1]}{k}$ and $\mathcal{B}_{n,k,s}=\{B\in \binom{[n]}{k}:B\cap [s-1]\neq \emptyset\}$. The case $s=2$ is simply the Erdős-Ko-Rado theorem on intersecting families and is well understood. The case $n=sk$ was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if $k,s$ are fixed and $n$ is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for $n\in [sk,sk+c_{s,k}]$ if $s$ is large enough compared to $k$. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most $s$ in a similar range of parameters. In this short note, we are concerned with the case $s\ge 3$ fixed, $k$ tending to infinity and $n\in\{sk,sk+1\}$. For $n=sk$, we show the stability of the unique extremal construction of size $\binom{sk-1}{k}=\frac{s-1}{s}\binom{sk}{k}$ with respect to minimal degree. As a consequence we derive $\lim\limits_{k\rightarrow \infty}\frac{f(sk+1,k,s)}{\binom{sk+1}{k}}<\frac{s-1}{s}-\varepsilon_s$ for some positive constant $\varepsilon_s$ which depends only on $s$.

math.CO↗