Combinatorics Papers of the Week

We construct infinite $B_h$ sets that repeatedly attain the density scale of finite $B_h$ sets.

Here we use the topology of the space of complex Hadamard matrices, which describe transitions between MUBs, to characterize all sets in dimension six.

We study connectivity, component geometry, and random walks throughout the regime $\log n\ll m<n/2$.

A direct application of the symmetric Lovász Local Lemma shows that $\textrm{HG}(G) < e\Delta$ for every graph $G$ of maximum degree at most $\Delta \ge 1$. We prove that this bound is asymptotically sharp: for every $\varepsilon > 0$ and every sufficiently large $\Delta$, there exists a graph $G$ of maximum degree at most $\Delta$ with $\textrm{HG}(G) \ge (e - \varepsilon)\Delta$.

We construct an explicit minimal cellular resolution for every squarefree monomial ideal generated in degree $n-2$ in the polynomial ring $S=K[x_1,\ldots,x_n]$.

We introduce the face polynomial of a ribbon graph, a generating function for face numbers under the ribbon group action.

In this paper, we investigate the generalized Turán number $\mathrm{ex}(Q^{k}_{n}, C_{2\ell})$ of $Q^{k}_{n}$ for all even integers $k\geq 4$ and integers $\ell\geq 2$, generalizing the result for $Q_{n}$ [J. Graph Theory 16 (1992) 273-286] from $k=2$ to $k\geq 4$.

In this paper, we investigate the generalized Turán number $\mathrm{ex}(G, C_{g})$ for graphs $G$ in which every edge is contained in a fixed number of $g$-cycles, and establish a general upper bound on it.

We determine it on $\ell_1$ products of betweenness-free metrics, those with no point between two others, a class containing every Hamming graph.

We prove that any linear $3$-query locally correctable code $C \colon \mathbb{F}_2^k \to \mathbb{F}_2^n$ with constant correction radius must have length $n \geq 2^{\Omega(\sqrt{k})}$.

Let $B_k$ denote the book consisting of $k$ triangles sharing a common edge. We prove that there exist absolute constants $\rho>0$ and $N$ such that, for every pair of integers $n,k$ with $n\ge N$ and $2\le k\le \rho n$, writing $n=5q+r$ with $0\le r<5$, \[ \operatorname{ex}(n,C_5,B_k)=q^{\,5-r}(q+1)^r, \] with the balanced complete $C_5$-blow-ups as the only extremal graphs.

We prove a fixed-characteristic density form of the Berger-Charpin conjecture for non-degenerate irreducible cyclic codes: for each prime $p$, the lengths admitting such a non-standard code over some finite field of characteristic $p$ have natural density zero.

In this paper, we prove their conjecture, showing that \[ \phi(v)\le\frac{c}{r}\frac{\log(d(v)+2)}{d(v)+1} \] suffices for an absolute constant $c>0$.

We give an $O_H(n^5)$-time algorithm to test whether an $n$-vertex labeled tour graph has a full immersion isomorphic to a labeled tour graph $H$.

We prove that every monadically dependent graph class has almost bounded merge-width, thereby confirming a conjecture of the first and the last author [STOC '25].

In particular, we show that in monadically dependent classes, tuples of fixed length have only $|U|^{O(1)}$ types of fixed quantifier rank over any set of parameters $U$.

We prove that the maximum universe size of a $t$-perfect hash family of length $N$ over an alphabet of size $q$ satisfies $ p_t(N,q)\ge q^{k-o(1)}$.

A conjecture of Frankl proposed in the 1980s states that for $n\ge2d$, every Sperner family $\mathcal{F}\subseteq 2^{[n]}$ of VC-dimension at most $d$ has at most $\binom{n}{d}$ members. We prove this conjecture for every fixed $d$ and large $n$, and show that equality holds only when $\mathcal{F}$ is $\binom{[n]}{d}$ or $\binom{[n]}{n-d}$.

In this paper, we study the graph shotgun assembly problem in the labeled lattice model, as introduced by Mossel and Ross (2019).

Motivated by the recent asymptotic formula for the moment of the alternating sum of parts in partitions by Craig and Li, we study the moment of the alternating sum of parts in compositions.