We prove an $\mathrm{MSO}_2$ zero-one law for a very sparse Erd\H{o}s-R\'enyi graph after pruning by component order. Let $p_n=c_n/n$, where $c_n\to0$, and delete every component of order less than $f(n)$, where $f(n)\to\infty$. If \[ f(n)\bigl(\log f(n)+\log(1/c_n)\bigr)=o(\log n), \] then the resulting graph satisfies a zero-one law for $\mathrm{MSO}_2$, with quantification over sets of vertices and sets of edges. The proof combines uniform component counts, an MSO Feferman-Vaught decomposition for disjoint unions, and semilinearity of the order spectra of MSO-definable classes of finite trees. We also show that the term $f(n)\log f(n)$ cannot simply be omitted: star components can occur at first-order-visible Poisson thresholds. We further establish first-order limit laws for bond percolation on the discrete torus $T_L^d$. In the two-sided subpolynomial regime, pruning below a sufficiently slow threshold yields a zero-one law. For the unpruned model in either one-sided polynomial regime, the reciprocal exponents $\alpha=1/k$ are precisely the critical scales. At such a scale, an extended limit of $N p_N^k$ or $N q_N^k$ equal to $0$ or $\infty$ gives a zero-one law; a positive finite limit gives a convergence law but not a zero-one law; and the absence of an extended limit gives failure of convergence. Finally, $\mathrm{MSO}_1$ already detects the parity of the torus side length through bipartiteness, producing a natural obstruction to monadic convergence in a near-deterministic regime.
For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erd\H{o}s and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)) \frac{n^2}{16}$. Balogh and Liu (JCTB, 2014) disproved this conjecture and determined the asymptotic value of $f_4(n,\mathrm{ex}(n,K_3)+1)$. He, Ma, Ma and Ye (JCTB, 2023) later determined $f_{p+1}(n,\mathrm{ex}(n,K_p)+1)$ asymptotically for every $p\ge 3$, and asked for the value of $f_{p+1}(n,m)$ for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. In this paper, we answer their question asymptotically for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. We also determine the exact value of $f_3(n,m)$ for all $0\le m\le \mathrm{ex}(n,K_3)$ by a different method.
We prove that for all fixed $k\geq 4$, any $N$ vertex graph with no independent set of size $n$ and $N\geq \Omega(n^{k-1}/\log^{k-2}n)$ contains at least $$ \Omega\bigg(\binom Nk \Big(\frac{\log n}{n}\Big)^{\binom k2}/\log n\bigg) $$ cliques of order $k$, and for $k\geq 5$ this is best possible conditional on the known upper bounds for $r(k,n)$. This is also true and tight for $k=2$ by Tur\'an's Theorem and for $k=3$ by a result of Bohman and Mubayi. We show the bound is also tight for $k=4$. We obtain other supersaturation results using the same methods.
For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1})=2m\left(\frac{n}{m}\right)^m+O_m\!\left(n^{m-1/5}\right).$$ The proof uses a sharp weighted cycle--path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.
Let $D_n$ be the disjointness graph on the nonempty subsets of $[n]$, whose independent sets are exactly the intersecting families on $[n]$. We study the weighted independent-set polynomial $W(n)=\sum_F\prod_{S\in F}w(S)$, the sum running over these families, for the doubly exponential weight $w(S)=2^{2^{n-|S|}}-1$. The kernel-bearing (trivial) part $Z_\cap(n)$ is exact by inclusion-exclusion and satisfies $Z_\cap(n)\sim n\cdot 2^{3^{n-1}}$. For the kernel-free remainder we prove the exact prefactor $R(n)=(3/4+o(1))n\cdot 2^{3^{n-1}-2^{n-1}+2}$, whence $\log_2(Z_\cap(n)/R(n))=2^{n-1}-2+\log_2(4/3)+o(1)$, an additive $o(1)$, not merely a leading-order one. The engine is a second-level extremal theorem: among kernel-free maximal linked systems other than the $n$ one-flip stars, the largest weight exponent is $3^{n-1}-3\cdot 2^{n-2}+6$, a fixed gap $2^{n-2}-4$ below the maximum, with the extremisers classified exactly. None of this is special to the weight: for $w_B(S)=B^{B^{n-|S|}}-1$ with integer $B\ge 2$ the same stars dominate, the near-extremal families sit a gap $B^{n-2}-B^2$ below, and the prefactor is $1-B^{-B}$. The combinatorial input is the $p$-biased extremal problem for non-trivial intersecting families: $M_2(n,p)=p-pq^{n-1}+qp^{n-1}$ for all $n\ge 3$, $0<p\le 1/2$, $q=1-p$. This first level is essentially known: the extremal family is the Wheel coterie of Peleg and Wool, and at $p=1/Q$ the statement, with its maximiser classification, is the case $r=n$ of Borg's Hilton-Milner theorem for signed sets (2013). We give a short self-contained Erd\H{o}s-Ko-Rado proof, uniform in real $p\in(0,1/2]$, whose layer-two rigidity feeds the second level. The novelty claimed lies at the second level and in the prefactor, where the classification cannot be read off the layer profile alone: at $n=5$ one profile carries two non-isomorphic types of extremisers.
Let $\zeta(G)$ denote the minimum number of parts in a partition of $V(G)$ in which every part induces either a clique or an independent set. Erd\H{o}s and Gimbel asked whether, for $G_n\sim G(n,1/2)$, the difference $\chi(G_n)-\zeta(G_n)$ tends to infinity with high probability. We resolve this problem along the full sequence $n\to\infty$ and prove that $\mathbb P(\chi(G_n)-\zeta(G_n)\ge ((\log 2)^2/4)\log(200/153)\,n/(\log n)^3)\to1$. This gives a lower bound at the conjectured scale $n/(\log n)^3$. We also obtain a phase-resolved refinement: if $\delta_n$ is the fractional part of the standard independence-number center, then the coefficient may be replaced by $(\log 2)^2A_4(\delta_n)/4-o(1)$, where $A_4$ is explicit, continuous, nonconstant, and satisfies $A_4(\delta)>\log(200/153)$ for every $\delta\in[0,1]$. The proof uses signed cocoloring profiles supported on four consecutive class sizes and remains uniform across jumps of the natural class-size cutoff. An exact signed-overlap identity separates local cell rewards from a binary cycle-space factor. A canonical decomposition into high cells and a capped residual matching, together with an endpoint-table comparison and an injective restriction of residual even edge sets, yields the required second-moment bound. A bounded-differences argument then amplifies the resulting rare signed witness to a high-probability cocoloring.
For fixed integers $s\ge t\ge2$, let $\operatorname{ex}(n,n,n,K_{s,t})$ denote the maximum number of edges in a tripartite $K_{s,t}$-free graph with $n$ vertices in each part. When $s\ge(t-1)!+1$, let $r$ be the largest integer satisfying $s\ge(t-1)!r^{t-1}+1$. Using the quotient norm graphs of Alon, R\'onyai and Szab\'o, we prove that \[ \operatorname{ex}(n,n,n,K_{s,t}) \ge \left(\frac{3}{2^{1/t}}r^{1-1/t}+o(1)\right)n^{2-1/t}. \] Improving an upper bound of Tait and Timmons, we prove that, for all $s\ge t\ge 2$, \[ \operatorname{ex}(n,n,n,K_{s,t})\le \left(\frac{3}{2^{1/t}}(s-t+1)^{1/t}+o(1)\right)n^{2-1/t}. \] Together, these bounds recover the results for $t=2$, and give the new asymptotic formula \[ \operatorname{ex}(n,n,n,K_{3,3}) =\left(\frac{3}{\sqrt[3]{2}}+o(1)\right)n^{5/3}. \] Analogous results extend to $k$-partite graphs containing no $K_{s, t}$ whose $s$-vertex or $t$-vertex side lies in a single part. As an application of our tripartite construction, we determine the tripartite multicolor Ramsey number of $K_{3,3}$ asymptotically.
Yantao Tang, Yi Zhao· 0 citations
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