arXiv subjects

Yaping Mao

Publications and source records attributed to Yaping Mao.

At least 19 recordsLinked to original sources

arXiv2026

Short permutation pattern waves: The complete length-five classification

For a permutation pi in Sk, a pi-wave is an increasing sequence of integers whose consecutive gaps have relative order pi. Let g(pi, n) be the maximum cardinality of a pi-wave-free subset of [n], and let P(pi, r) be the least integer N such that every r-coloring of [N] contains a monochromatic pi-wave. We determine the asymptotic orders of both functions for every permutation of length at most five. Existing upper bounds and weak-wave constructions provide sharp bounds for all but eight length-five patterns in S_5. For these eight exceptional alternating patterns, we establish g(pi, n) = Theta((log n)squared) and P(pi, r) = Theta(r(log r)squared).

math.CO↗

arXiv2026

A Solution to the Kohayakawa-Prömel-Rödl Conjecture

The induced Ramsey number $r_{\mathrm{ind}}(G,H)$ is defined as the minimum order of a graph $F$ such that any 2-coloring of its edges with red and blue leads to either a red induced copy of $G$ or a blue induced copy of $H$. In 1998, Kohayakawa, Prömel, and Rödl conjectured that for any fixed graph $G$, there exists a constant $f=f(G)$ such that the induced Ramsey number $r_{\mathrm{ind}}(G, H)$ is bounded by $|V(H)|^f$ for every graph $H$. In this paper, we completely solve the Kohayakawa--Prömel--Rödl conjecture, which states that for any graphs $G$ and $H$ with $|V(G)|=k\ge2$ and $|V(H)|=t\ge2$, $r_{\mathrm{ind}}(G,H)\le t^{60k^2+3}$. This verifies the conjecture with $f(G)=60|V(G)|^2+3$.

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arXiv2026

Close Divisors of Typical Integers:The Ford--Green--Koukoulopoulos Conjecture

For an integer $k\geq2$, let $α_k$ be the supremum of the real numbers $a$ for which almost every integer $n\geq2$ has divisors $d_1<\cdots β_k/(1-β_k)$, almost every integer $n\geq2$ has no divisors $d_1<\cdots<d_k\mid n$ satisfying $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr)$. We also correct local errors in their paper [\emph{Invent. Math.} 232 (2023), 1027--1160], concerning the finite-subflag reduction, the residual-sum count, the moment estimate and the lattice adjustment. These corrections preserve the entropy-threshold comparison used in our proof.

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arXiv2026

Multicolor Ramsey and list Ramsey numbers for star-like trees

For a graph $H$, the $k$-color Ramsey number $r(H;k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $H$. A $k$-list assignment has $|L(e)|=k$ for every edge. The list Ramsey number $r_\ell(H;k)$ is the least integer $N$ for which there exists a $k$-list assignment on $E(K_N)$ such that every coloring from the lists contains a monochromatic copy of $H$. Let $K_{1,n}$ be a star, $S(n,m)$ the double star obtained by joining the centers of $K_{1,n}$ and $K_{1,m}$, and $S_n^m$ the graph obtained from $K_{1,n}$ by subdividing $m$ edges once. Alon et al.\ conjectured that $r_\ell(K_{1,n};k)=r(K_{1,n};k)$ for all $k,n\ge1$. In this paper, we confirm their conjecture for all $k\ge1$ and $n\ge3$ by a unified direct proof. For even $k\ge4$ and under explicit parameter conditions, we prove that $r(S(n,m);k)=kn+m+2$ for even $n$ and $r(S_n^m;k)=k(n-1)+m+2$ for odd $n$. For two colors, we establish a general list Ramsey lower bound and determine the common values of Ramsey and list Ramsey numbers for double and subdivided stars in explicit parameter ranges. These results close several gaps in the known bounds.

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arXiv2026

Dense ascending waves: A resolution of the Alon-Spencer conjecture

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. A strictly increasing sequence of integers $x_1<\cdots<x_k$ is an \emph{ascending wave} if its consecutive differences are nondecreasing. Let $g(n)$ be the largest integer $k$ such that every set $A\subseteq[n]$ with $|A|\ge n/2$ contains an ascending wave of length $k$. Alon and Spencer proved that \[ c_1\frac{(\log n)^2}{\log\log n}\le g(n)\le c_2(\log n)^2 \] for all sufficiently large $n$, and they conjectured that the factor $\log\log n$ in the lower bound can be removed. In this paper, we confirm their conjecture.

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arXiv2026

Constructing edge-disjoint Steiner trees in Cartesian product networks

Cartesian product networks are always regarded as a tool for ``combining'' two given networks with established properties to obtain a new one that inherits properties from both. For a graph $F=(V,E)$ and a set $S\subseteq V(F)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a subgraph $T=(V',E')$ of $F$ that is a tree with $S\subseteq V'$. For $S\subseteq V(F)$ and $|S|\geq 2$, the {\it generalized local edge-connectivity} $λ(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $F$. For an integer $k$ with $2\leq k\leq n$, the {\it generalized $k$-edge-connectivity} $λ_k(F)$ of a graph $F$ is defined as $λ_k(F)=\min\{λ(S)\,|\,S\subseteq V(F) \ and \ |S|=k\}$.In this paper, we give sharp upper and lower bounds for $λ_k(G\Box H)$, where $\Box$ is the Cartesian product operation, and $G,H$ are two graphs.

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arXiv2026

Gap spectra and densities of slow Fibonacci walks

Let $F_1=F_2=1$ and $F_{t+2}=F_{t+1}+F_t$ for $t\geq1$. For every $n\geq2$, there are unique integers $a,b,t$ such that $n=aF_t+bF_{t-1}$ with $t\geq2$ and $1\leq a\leq b\leq F_t$. The Fibonacci walk with initial pair $(b,a)$ reaches $n$ as late as possible, and the term following $n$ in this walk is $\lfloorϕn\rfloor$ when $t$ is even and $\lceilϕn\rceil$ when $t$ is odd, where $ϕ=(1+\sqrt5)/2$. Let $D=\{d_1<d_2<\cdots\}$ and $U=\{u_1<u_2<\cdots\}$ be the sets corresponding to even and odd $t$, respectively. For $\ell,m\geq1$, define $D_\ell=\{d_{k+\ell}-d_k:k\geq1\}$, $U_\ell=\{u_{k+\ell}-u_k:k\geq1\}$, $D_\ell(m)=\{d_k:d_{k+\ell}-d_k=m\}$ and $U_\ell(m)=\{u_k:u_{k+\ell}-u_k=m\}$. Chung, Graham and Spiro conjectured that $D_\ell=U_\ell$ for all $\ell$, and asked for the densities of $D_\ell(m)$ and $U_\ell(m)$, especially when $\ell=1$. In this paper, we determine the third and fourth order gap spectra, and show that the conjecture holds for $\ell=3$ but fails for $\ell=4$. We also answer their density question by characterizing when $D_\ell(m)$ and $U_\ell(m)$ have natural densities and proving that their logarithmic densities always exist and are equal. For $\ell=1$, we give the exact logarithmic densities.

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arXiv2026

New upper bound for multicolor Ramsey numbers

Let $R_r(k)$ denote the diagonal $r$-color graph Ramsey number. We prove that there exist absolute constants $c,K>0$ such that \[ R_r(k)\le \exp\!\left(-c\frac{k}{r^2\log^4(2r)}\right)r^{rk} \] for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. The proof combines a positive-coefficient root filter of variable order with a retained-spine refinement of the multicolor book method.b

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arXiv2026

Ramsey multiplicity for ordered graphs

Let $\cG_1,\ldots,\cG_k$ be fixed vertex-ordered graphs, each containing at least one edge. The ordered Ramsey number $\oR(\cG_1,\ldots,\cG_k)$ is the least integer $N$ such that every $k$-edge-coloring of the ordered complete graph $\cK_N$ contains an order-preserving copy of $\cG_i$ in color $i$ for some $i\in[k]$. For positive weights $\blambda=(λ_1,\ldots,λ_k)$, let $\oM_{\blambda}(n;\cG_1,\ldots,\cG_k)$ denote the minimum weighted number of correctly colored, order-preserving copies of the target graphs over all $k$-edge-colorings of $\cK_n$. When $\blambda=\bf{1}$, $\oM_{\bf{1}}(n;\cG_1,\ldots,\cG_k)=\oM(n;\cG_1,\ldots,\cG_k)$ is called the ordered Ramsey multiplicity. In this paper, we first establish the amplification inequality \[ \oM_{\blambda}(n;\cG_1,\ldots,\cG_k) \ge \oM_{\blambda}(t;\cG_1,\ldots,\cG_k) \frac{\binom{n}{\hmin}}{\binom{t}{\hmin}}, \] where $h_i=v(\cG_i),\hmin=\min_{i\in[k]}h_i$, and $n\ge t\ge\oR(\cG_1,\ldots,\cG_k)$. Let $\cS_{r,s}$ be the ordered star whose center has $r-1$ leaves to its left and $s-1$ leaves to its right, and let $\bB_m$ be the family of all ordered perfect matchings on $[2m]$ containing the edge $\{1,2m\}$. We apply the amplification inequality to obtain the multiplicity lower bounds for ordered stars and ordered perfect matchings. We then obtain the upper bound $\oM_{\boldsymbolλ} (n;\cS_{r_1,s_1},\cS_{r_2,s_2}) \le \min\{λ_1 B_{h_1}(n),λ_2 B_{h_2}(n)\}$ by constructions, where $B_{h_i}(n):= \binom{\lfloor n/2\rfloor}{h_i} + \binom{\lceil n/2\rceil}{h_i}$ and $h_i=r_i+s_i-1$ for $i\in [2]$. We also derive a random-coloring upper bound for ordered stars and prove \[\oM(n; \bB_m,\bB_m) \le \binom{n}{2m} \frac{(2m-2)!}{2^{2m-2}(m-1)!}.\] Finally, we establish a regularity-based lifting theorem for ordered colorings.

math.CO↗

arXiv2026

The inversion number of a path-reversed tournament: Resolving a conjecture of Belkhechine, Bouaziz, Boudabbous, and Pouzet

Let $D$ be a tournament and let $X\subseteq V(D)$. The inversion of $X$ reverses all arcs whose both endpoints lie in $X$ and leaves every other arc unchanged. A family of inversions is a decycling family if applying all of them produces an acyclic, equivalently transitive, tournament. The inversion number $\inv(D)$ is the minimum size of such a family. Let $Q_n$ be the tournament on $[n]$ obtained from the natural transitive tournament by reversing precisely the consecutive pairs $12,23,\ldots,(n-1)n$. Belkhechine, Bouaziz, Boudabbous, and Pouzet conjectured in their unpublished manuscript that a natural path-reversed family has inversion number exactly $\left\lfloor(n-1)/2\right\rfloor$. The same problem was later recorded by Bang-Jensen, da Silva, and Havet and by Alon, Powierski, Savery, Scott, and Wilmer. In this paper we resolve this conjecture.

math.CO↗

arXiv2026

Solution to a conjecture of Alon, Dębski, Grytczuk and Przybyło on fixed-cardinality arithmetic progressions

Fix a positive integer $n$, and put $B_d=\{d,2d,\ldots,nd\}$. Let $M_k(n)$ be the least integer $m$ for which one translate of each of $B_1,\ldots,B_k$ can be placed pairwise disjointly in $[m]$. We prove that, for every $\eps\in(0,1)$ and all sufficiently large $k$, one has $M_k(n)\le n\lceil(1+\eps)k\rceil$. Since the trivial counting bound gives $M_k(n)\ge nk$, it follows that $M_k(n)=(1+o(1))nk$ for every fixed $n$. This confirms a conjecture of Alon, Dębski, Grytczuk and Przybyło on prescribed-difference packings of fixed-cardinality arithmetic progressions.

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arXiv2026

Multiplicity for partially ordered sets

Let $\mathcal Q=\{Q_a:a\geq1\}$ be a nested family of finite posets such that $Q_a\subseteq Q_{a+1}$ and $|Q_a|<|Q_{a+1}|$. For a poset $Q$, let $\mathcal C_t(Q)$ denote the set of all strict $t$-chains in $Q$. Given an $r$-coloring of $\mathcal C_t(Q_a)$ and posets $P_1,\ldots,P_r$, a weak copy of $P_i$ is called monochromatic of color $i$ if all $t$-chains in the copy have color $i$; the strong version is defined in the same way for induced copies. The corresponding weak and strong multiplicity parameters are the minimum possible total number of such monochromatic copies in the host poset.For the Boolean lattice $B_n$, define $E_n={(S,T,U)\in B_n^3:S\subsetneq T\subsetneq U,\ |S|+|T|=|U|}.$ For a two-coloring $χ:B_n\to{0,1}$, a triple $(S,T,U)\in E_n$ is monochromatic if $χ(S)=χ(T)=χ(U)$. Let $R^{\mathrm{arith}}_2$ be the least integer $n$ such that every two-coloring of $B_n$ contains a monochromatic triple in $E_n$, and let $M^{\mathrm{arith}}_2(B_n)$ be the minimum number of monochromatic triples in $E_n$ over all two-colorings of $B_n$. We prove that $R^{\mathrm{arith}}_2=9.$ Moreover, $|E_n|=\binom{2n}{n}-[x^n](1+x+x^2)^n-2^n+1=\frac{4^n}{\sqrt{πn}}\bigl(1+o(1)\bigr),$ and $2^{δn+o(n)}\le M^{\mathrm{arith}}_2(B_n)\le 2^{γn+o(n)}, $ where $δ\approx 1.356779$ and $γ\approx 1.567837$ are explicit entropy constants. For general nested host families, we prove a double-counting lower bound for strong poset multiplicity. For an arbitrary finite host poset $R$, we also introduce a Fourier-Möbius method and give an exact Fourier expansion for strong multiplicity, a Parseval-type error bound, and a spectral lower bound.

math.CO↗

arXiv2026

A complete solution to the biased Alon-Krivelevich-Spencer-Szabó criterion problem for the discrepancy game

Let $H=(V,\mathcal E)$ be a finite hypergraph. For positive integers $p$ and $q$, the $(p:q)$-biased discrepancy game on $H$ is played in complete rounds. In each round, Balancer first claims $p$ previously unclaimed vertices, and then Unbalancer claims $q$ previously unclaimed vertices. Let $B$ and $U$ be the final sets of vertices claimed by Balancer and Unbalancer, respectively. For an edge $e\in\mathcal E$, define $D_e = q|B\cap e|-p|U\cap e| = (p+q)|B\cap e|-p|e|$. Thus $D_e$ measures the deviation of Balancer's share of $e$ from the density $p/(p+q)$. In 2005, Alon, Krivelevich, Spencer and Szabó proved a Chernoff-type potential criterion for the unbiased alternating discrepancy game, corresponding to the case $p=q=1$, and asked for a biased analogue for general $p,q$. In this paper, we prove a complete biased analogue in the complete-round formulation. More precisely, for every finite hypergraph $H=(V,\mathcal E)$ and every fixed bias $(p:q)$, we give an explicit exponential condition under which Balancer has a strategy forcing $-L_e^- \le D_e \le L_e^+$ for every $e\in\mathcal E$, where $L_e^+$ and $L_e^-$ are prescribed edge-dependent target values.

math.CO↗

arXiv2026

Ramsey-Turán theory for partially-ordered sets

We introduce weak and strong poset Ramsey-Turán numbers for $t$-chains in host poset families, focusing on the Boolean lattice family $\mathcal{B}=\{B_n:n\ge 1\}$. For any poset $P$, we show $\operatorname{RT}(\mathcal{B};n,P,l,t)\le \operatorname{RT}^{\sharp}(\mathcal{B};n,P,l,t)$, with equality when $P$ is a chain. In particular, for $t=1$, $\operatorname{RT}(\mathcal{B};n,C_k,l)=\operatorname{RT}^{\sharp}(\mathcal{B};n,C_k,l)=(k-1)(l-1)$. We also give universal upper bounds for both versions. For fixed $k,l,t$ with $\min\{l-1,k-1\}\ge 1$, we prove $\operatorname{RT}^{\sharp}(\mathcal{B};n,A_k,l,t)=Θ(n^t)$. More generally, for every non-chain poset $P$, the strong number is $Θ(n^t)$ for fixed $l,t$. Finally, if $h(P)=r>t$ and $l(n)=\lfloor M_n^β\rfloor$ with $0<β\le α<1$, then both weak and strong versions admit lower bounds of order $Ω\!\left(2^{βn}n^{-β/2}\right)$.

math.CO↗

arXiv2026

Diagonal Ramsey numbers for wheels

The Ramsey number $\mathrm{R}(G_1,G_2)$ is the smallest integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ contains either a red copy of $G_1$ or a blue copy of $G_2$. In 2022, the third author and others gave lower and upper bounds of the Ramsey number $\mathrm{R}(W_n,W_n)$, where $W_n$ is the wheel graph with $n$ vertices. In this paper, we improve their bounds by showing that $3n-2\leq \mathrm{R}(W_n,W_n)\leq 6n-6$ for even $n\geq 8$ and $2n\leq \mathrm{R}(W_n,W_n)\leq \frac{9n-7}{2}$ for odd $n\geq 7$. Furthermore, we give recursive bounds for the $k$-colored Ramsey number for $W_n$.

math.CO↗

arXiv2026

From Halin's Edge Removability to Matching Removability in $k$-Connected Graphs

We study matching-removability under the degree/connectivity regime of Halin's theorem, which asserts that every $k$-connected graph $G$ with minimum degree $δ(G)\ge k+1$ contains an edge $e$ such that $G-e$ remains $k$-connected. For $k,\ell\ge 1$, an $\ell$-matching is a matching of size $\ell$. A matching $M$ in a $k$-connected graph $G$ is {\it $k$-removable} if $G-M$ remains $k$-connected. We improve Halin's result by proving that every $k$-connected graph $G$ with $δ(G)\ge k+1$ contains a $k$-removable $2$-matching, except when $k=1$ and $G$ is a cycle. For small $k$ we obtain stronger bounds: (i) $k=1$: a 1-removable $\min\{\lfloor n/2\rfloor,δ(G)\}$-matching; (ii) $k=2$: a 2-removable $\lceil(δ(G)+1)/2\rceil$-matching, with a unique tight exception when $δ(G)$ is even and $G\cong K_{δ(G)+1}$; and (iii) $k=3$: for $δ(G)\ge 5$, a $3$-removable $\lceil(δ(G)+1)/2\rceil$-matching. All these bounds are optimal with respect to removable matching size and minimum degree. We also show that for every $n\ge 2δ$, there exists a $k$-connected $n$-vertex graph $G$ with minimum degree $δ$ that does not contain a $k$-removable matching of size at least $δ(G)+1$. Moreover, for $k\le 2$ there exists a $k$-removable $(δ(G)-c)$-matching for some $c\le 3$, which is optimal up to the additive constant.

math.CO↗

arXiv2026

Erdős-Gyárfás problem for partially ordered sets

Given integers $p,q,t$ with $1 \le t \le p$ and $1 \le q \le h_p(t)$, a strong $(p,q,t)$-coloring of the Boolean lattice $B_n$ is a coloring of its $t$-chains such that every induced copy of $B_p$ in $B_n$ uses at least $q$ colors on its $t$-chains. Let $f_t^{\sharp}(n,p,q)$ denote the minimum number of colors in such a coloring. We study this Boolean-lattice analogue of the Erdős-Gyárfás function.We first show that every finite poset strongly embeds into a Boolean lattice. Combined with a structural Ramsey theorem for finite posets with linear extensions, this implies the existence of the strong Boolean Ramsey number $\mathrm{R}^{\sharp}_{k,t}(\mathcal{B}\mid Q)$ for every integer $k\ge1$, every $t\ge1$, and every nonempty finite poset $Q$. In particular, this gives an affirmative answer to a problem of Cox and Stolee and yields the existence of $f_t^{\sharp}(n,p,2)$. Next, using the symmetric Lovász local lemma, we obtain a probabilistic upper bound on $f_t^{\sharp}(n,p,q)$. Finally, we prove lower bounds by combining Turán-type extremal estimates for $t$-chains, a double-counting argument, and a generalized Lubell-type framework for $t$-chains.

math.CO↗

arXiv2026

Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erdős-Faudree-Schelp conjectures

For finite graphs $G$ and $H$, let $\RR(G,H)$ denote the isomorphism classes of Ramsey-minimal graphs for $(G,H)$. We prove two 1981 conjectures of Burr, Erdős, Faudree, Rousseau, and Schelp: Ramsey-finiteness is preserved by adjoining disjoint matchings, and $(G,H)$ is Ramsey-infinite unless both graphs are odd stars or one graph has a $K_2$ component. We also replace Burr's stronger 1979 survey characterization by the correct necessary-and-sufficient form: apart from the matching case and the odd-star-with-matchings case, the only additional finite pairs are Faudree's star-forest family.

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