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Cosmin Pohoata

Publications and source records attributed to Cosmin Pohoata.

At least 19 recordsLinked to original sources
arXiv2026

The sharp exponent for the minimal distance problem

We show that for every fixed $\varepsilon>0$, there exist arbitrarily large families of point-line pairs $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$, with $x_i \in \ell_i$ for all $i$, and such that $\operatorname{dist}(x_i,\ell_j)\ge n^{-2/3-\varepsilon}$ for all $i \neq j$. Combined with a previous result of Cohen, the author, and Zakharov, this solves the minimal distance problem. The same construction also comes with an unexpected finite field consequence: for every $\varepsilon>0$, there exists a set of primes $q$ of positive relative density for which $\mathbb F_q^2$ contains an induced point-line matching of size $\gtrsim q^{3/2-\varepsilon}$. This disproves a conjecture of Hunter, the author, Verstra\"ete and Zhang.

math.CO↗
arXiv2026

The Minkowski grid has robustly many repeated distances

We show that there exists a constant $\delta > 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \subset \mathbb{R}^2$ with the following property: for every subset $A \subseteq P$ of size $|A| \geq 2$, \[ \max_{\lambda>0} \#\{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=\lambda\} \gtrsim \frac{|A|^2}{n^{1-\delta}}.\] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erd\H{o}s unit distance problem and the Elekes-R\'onyai problem. Taking $A=P$, the inequality above gives a distance occurring $n^{1+\delta}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture. In addition, the same inequality shows that (1) all subsets of $P$ of size $\gtrsim n^{1-\delta}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-\delta}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erd\H{o}s. The existence of a set satisfying property (1) confirms a conjecture of Erd\H{o}s from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.

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arXiv2026

A combinatorial large sieve for Sidon sets, distances, and norm forms

We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset $A\subset\{1^2,\ldots,N^2\}$ satisfies \[ |A| \le N\exp\left( -c\frac{\log N}{\log\log N} \right) \] for some absolute constant $c>0$. This gives the first super-polylogarithmic saving for a classical problem of Alon and Erd\H{o}s. As a second application, we establish new upper bounds for two grid-distance problems. We show that the largest subset of $[N]^2$ with no repeated distance has size at most $N\exp\left(-c\log N/\log\log N\right)$, giving the first progress in over thirty years on a problem of Erd\H{o}s and Guy. The same method also gives a similar saving for subsets of $[N]^2$ with no isosceles triangles, a problem recently popularized by Ellenberg and by the PatternBoost work of Charton, Ellenberg, Wagner, and Williamson. We then develop an entropic version of the method. This gives bounds for $B_2[g]$-sets in the squares and for analogous bounded-multiplicity problems associated with norm forms over arbitrary number fields. More importantly, this new method also allows us to establish the first nontrivial bounds for $B_3[g]$-sets in the cubes and $B_4[g]$-sets in the fourth powers.

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arXiv2026

Split primes and the Elekes-R\'onyai problem

There exist an absolute constant $c>0$ and arbitrarily large finite sets $A\subset \mathbb{R}$ with $$\left| \left\{x+y+(x-y)^2:\ x, y \in A\right\}\right| \le|A|^{2-c}.$$ Since $x+y+(x-y)^2 \in \mathbb{R}[x,y]$ is a polynomial which is neither additive nor multiplicative, this provides a counterexample for the Elekes-R\'onyai problem. The proof combines two amplifications of the same local congruence defect: horizontal amplification over squarefree products of rational primes, and vertical amplification through bounded root-discriminant towers in which those primes split completely. In this way a fixed local density defect becomes macroscopic, producing a power saving. This phenomenon also suggests a broader mechanism for producing similar extremal constructions throughout combinatorics and number theory.

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arXiv2026

Above and below

We study a family of above-below Ramsey functions $\operatorname{AB}^{(d)}(k)$ defined for sequences of points in $\mathbb R^d$ whose projections to $\mathbb R^{d-1}$ have cyclic order type. The case $d=3$ is the above-below function $\operatorname{AB}(k)$ that was first introduced by Pohoata and Zakharov in their work on the Erd\H{o}s-Szekeres problem in $\mathbb{R}^{3}$. We prove the sharp estimate \[ \operatorname{AB}(k)=2^{2^{\Theta(k)}}, \] and, more generally, show that $\operatorname{AB}^{(d)}(k)$ is closely related to the higher-order cup-cap function of Eli\'a\v{s} and Matou\v{s}ek and the monotone Ramsey numbers of Balko.

math.CO↗
arXiv2026

$K_{2,t+1}$-free graphs with many copies of $K_{t,t}$

For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[ \mathrm{ex}(n,K_{t,t},K_{2,t+1})=\Theta_t(n^2). \] This answers a question of Spiro.

math.CO↗
arXiv2026

Improved Ramsey bounds for generalized Schur equations

We show that for $m, r \in \mathbb{N}$ and $N > (2m+1)^r (r!)^{1/m}$, every $r$-coloring of the integers in the interval $[N]$ contains a monochromatic solution to the equation \[ x_1 + \dots + \dots x_{m+1} = y_1 + \dots + y_m. \] This generalizes and improves recent results of Ko\'scuiszko. We also show that if $N \geq 2^{r}$, then every $r$-coloring of the integers in $[N]$ must always determine a monochromatic solution to the above equation for some $m \geq 1$. The latter estimate is optimal.

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arXiv2026

Two counterexamples to a conjecture about even cycles

A conjecture of Verstra\"ete states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by K\"uhn and Osthus in 2004. We identify two counterexamples to this conjecture for $\ell = 4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger's construction for extremal $C_{10}$-free graphs.

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arXiv2026

Grid-free linear hypergraphs via Cayley-Bacharach

We give a new construction showing that for every $r\ge 3$, there exists an $r$-uniform linear hypergraph on $n$ vertices with $\Theta_r(n^2)$ edges and no copy of the $r\times r$ grid. This complements the works of F\"uredi--Ruszink\'o, Glock--Joos--Kim--K\"uhn--Lichev, Delcourt--Postle for $r \geq 4$, as well as the subsequent constructions of Gishboliner--Shapira and Solymosi for the case $r=3$.

math.CO↗
arXiv2026

Large point-line matchings and small Nikodym sets

For any integer $d \geq 2$ and prime power $q$, we construct unexpectedly large induced matchings in the point-line incidence graph of $\mathbb{F}_{q}^{d}$ by leveraging a new connection with the Furstenberg-S\'ark\"ozy problem from arithmetic combinatorics. In particular, we significantly improve the previously well-known baselines when $q$ is prime, showing that $\mathbb{F}_{q}^{2}$ contains matchings of size $q^{1.233}$ and $\mathbb{F}_{q}^{d}$ contains matchings of size $q^{d-o_{d}(1)}$. These results and their proofs have several applications. First, we also obtain new constructions for finite field Nikodym sets in dimension $d \geq 2$, improving recent results of Tao by polynomial factors. For example, when $q$ is prime, we show the existence of Nikodym sets in $\mathbb{F}_q^d$ of size $q^d - q^{d - o_d(1)}$. Second, we construct a new minimal blocking set in $\mathrm{PG}(2,q)$, solving a longstanding problem in finite geometry. Third, we obtain new constructions for the minimal distance problem (in $\mathbb{R}^{2}$ and also in higher dimensions), improving a recent result of Logunov-Zakharov. We also obtain analogous results for general finite fields with large characteristics. In particular, in one of our constructions we introduce a new special set of points inside the norm hypersurface in $\mathbb{F}_{q}^{d}$, which directly generalizes the classical Hermitian unital and which may be of independent interest for applications.

math.CO↗
arXiv2026

Roth-type theorems in $K_{s,t}$-free sets

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $\Omega(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets. We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lov\'asz-Sauermann.

math.CO↗
arXiv2026

An update on multicolor Ramsey lower bounds

Building upon previous works by Conlon-Ferber and Wigderson, Sawin showed a few years ago that upper bounds on the minimum density of independent sets in a $K_t$-free $G$ can be used to provide lower bounds for multicolor Ramsey numbers. In this note, we observe how a further improved upper bound on this parameter directly follows from a recent spherical random geometric graph construction of Ma-Shen-Xie. As a consequence, we derive a small exponential improvement over the best known lower bounds for multicolor Ramsey numbers.

math.CO↗
arXiv2026

Distinct permutation dot products

We show that for any two sets of reals numbers $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_n\}$, the sums of the form $\sum_{i=1}^n a_i\,b_{\pi(i)}$ always take on $\Omega(n^{3})$ distinct values, as we range over all permutations $\pi \in S_n$. An important ingredient is a ``supportive'' version of Hal\'asz's anticoncentration theorem from Littlewood-Offord theory, which may be of independent interest.

math.CO↗
arXiv2026

A Hal\'asz-type theorem for permutation anticoncentration

Given a set $A=\{a_1,\ldots,a_n\}$ of real numbers and real coefficients $b_1,\ldots,b_n$, consider the distribution of the sum obtained by pairing the $a_i$'s with the $b_i$'s according to a uniformly random permutation. A recent theorem of Pawlowski shows that as soon as the coefficients are not all equal, this distribution is always spread out at scale $n^{-1}$: no single value can occur with probability larger than $\frac{1}{2\lceil n/2\rceil + 1}$, and this bound is sharp in general. We show that stronger anticoncentration holds when the coefficients have additional diversity. We quantify the structure of the coefficient multiset by a simple statistic depending on its multiplicity profile, and prove that the maximum point mass of the permuted sum decays polynomially faster as this statistic grows. In particular, when the coefficients are all distinct we obtain a bound of $n^{-5/2+o(1)}$, which can be regarded as an analogue of a classical theorem of Erd\H{o}s and Moser.

math.CO↗
arXiv2025

Chromatic Polynomial Evaluation Spectra

Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials $P_{G}$ arising from graphs $G$ on $n$ vertices grows exponentially with $n$, by establishing that the (dual) flow polynomial $F_{G}\left(\frac{3+\sqrt{5}}{2}\right)$ already takes on exponentially many values, if one varies $G$ over all planar cubic graphs $G$ on $n$ vertices. We show, more generally, that the size of the set $\{P_G(q): |V(G)|=n\}$ is exponential in $n$, for every fixed real number $q \neq 0,1,2$. In fact, our approach can also be pushed to show that $P_{G}(q)$ already takes on exponentially many values, if we only vary $G$ over all planar graphs on $n$ vertices. The case $q=3$ confirms a conjecture of Agol, which was initially motivated by the $\mathsf{NP}$-completeness of planar $3$-colorability.

math.CO↗
arXiv2025

On exponential Freiman dimension

The exponential Freiman dimension of a finite set $A \subset \mathbb{R}^{m}$, introduced by Green and Tao in 2006, represents the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped. For every $d \geq 1$, we precisely determine the largest constant $C_{d}>0$ (exponential in $d$) for which $$|A+A| \geq C_{d}|A| - O_{d}(1)$$ holds for all sets $A$ with exponential Freiman dimension $d$.

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