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István Tomon

Publications and source records attributed to István Tomon.

At least 19 recordsLinked to original sources
arXiv2026

Tight bounds for positive discrepancy via eigenvalues

Given an $n\times n$ symmetric matrix $M$ with largest eigenvalue $λ_1\geq 0$, it is easy to show that the solution of the optimisation problem $\max_{v\in [-1,1]^n}v^TMv$ is at most $λ_1 n$. We prove the following converse: if every $n'\times n'$ principal submatrix of $M$ has maximal eigenvalue at least $λ$, then $\max_{v\in [-1,1]^n}v^TMv\geq λ(n-n'+1)$. This allows us to lift results about the second largest and the smallest eigenvalue of graphs to the (a priori more difficult) setting of positive discrepancy and MaxCut. Using this new connection, we prove that every $n$-vertex $m$-edge graph that is far from a disjoint union of cliques has a cut of size at least $m/2+n^{5/4-o(1)}$, which is sharp up to the $o(1)$-term. Moreover, we prove that every $d$-regular $n$-vertex graph has bisection width at most $dn/4-Ω_{\varepsilon}(d^{1/3}n)$ for $d\leq (1-\varepsilon)n/2$, which is optimal for $d=Ω(n)$. This confirms a conjecture of Räty, Sudakov and Tomon.

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arXiv2026

Cross-free families have linear size

Two subsets $A$ and $B$ of a ground set $X$ are crossing if none of the four sets $A\setminus B,B\setminus A,A\cap B, X\setminus (A\cup B)$ are empty. Almost fifty years ago, Karzanov and Lomonosov conjectured that every family of subsets of an $n$-element ground set with no $k$-pairwise crossing members has size $O(kn)$. We prove the bound $O_k(n)$, settling (arguably) the main problem about the growth rate of such families. In fact, we prove the stronger statement that every $k$-wise laminar family has at most linear size. Here a family is $k$-wise laminar if it contains no antichain of size $k$ whose members have nonempty common intersection.

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arXiv2026

Box-Delaunay graphs of large chromatic number

For every $n$, we construct a 2-dimensional $n$-element poset whose Hasse diagram has independence number $n\exp(-Ω(\sqrt{\log n}))$, and consequently, chromatic number $\exp(Ω(\sqrt{\log n}))$. This also yields an $n$-point planar set whose box-Delaunay graph satisfies the same bounds.

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arXiv2026

Discrepancy of geometric incidences

We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degree at most $k$ has discrepancy at most $n^{\frac12-\frac{1}{2(D+1)}-\varepsilon}$ for some $\varepsilon=\varepsilon(D,k)>0$. This gives a polynomial improvement over the straightforward VC-dimension bound $\tilde O(n^{\frac12-\frac{1}{2(D+1)}})$. In the opposite direction, we construct $n$-point sets in $\mathbb R^d$ whose discrepancy with respect to hyperplanes is $\tildeΩ(n^{\frac12-\frac{1}{d+1}}),$ extending the point-line discrepancy lower bound of Chazelle and Lvov. We present further applications of our methods in communication complexity, concerning separation between randomized communication cost and deterministic communication cost with access to equality oracle.

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arXiv2026

Explicit constructions of optimal blocking sets and minimal codes

A strong $s$-blocking set in a projective space is a set of points that intersects each codimension-$s$ subspace in a spanning set of the subspace. We present an explicit construction of such sets in a $(k - 1)$-dimensional projective space over $\mathbb{F}_q$ of size $O_s(q^s k)$, which is optimal up to the constant factor depending on $s$. This also yields an optimal explicit construction of affine blocking sets in $\mathbb{F}_q^k$ with respect to codimension-$(s+1)$ affine subspaces, and of $s$-minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong $1$-blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong $s$-blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest.

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arXiv2026

Random 0/1-polytopes expand rapidly

A 0/1-polytope is the convex hull of a subset $V\subseteq \{0,1\}^n$. A celebrated conjecture of Mihail and Vazirani asserts that the graph of every 0/1-polytope has edge-expansion at least 1. In this paper, we show that typical 0/1-polytopes have significantly stronger expansion. Specifically, if $V$ is formed by sampling each vertex of $\{0,1\}^n$ independently with constant probability $p$, then with high probability the edge-expansion is $Θ(n)$ for $p \in (1/2, 1)$, and $n^{Θ(\log \log n)}$ for $p \in (0, 1/2)$. This improves the previously best known bound $Ω(1)$ due to Ferber, Krivelevich, Sales and Samotij.

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arXiv2026

Ramsey theory of low-degree semialgebraic relations

We prove that hypergraphs defined by low-degree polynomial inequalities contain large homogeneous subsets. Formally, let $H$ be an $r$-uniform hypergraph on $N$ vertices that is semialgebraic of constant description complexity, and each defining polynomial has degree at most $D$. Then $H$ contains a clique or an independent set of size $n$, where $N\leq \mbox{tw}_{3D^3}(n)$.

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arXiv2025

From small eigenvalues to large cuts, and Chowla's cosine problem

We prove that every graph with average degree $d$ and smallest adjacency eigenvalue $|λ_n|\leq d^γ$ contains a clique of size $d^{1-O(γ)}$. A simple corollary of this yields the first polynomial bound for Chowla's cosine problem (1965): for every finite set $A\subseteq \mathbb{Z}_{>0}$, the minimum of the cosine polynomial satisfies $$\min_{x\in [0, 2π]}\sum_{a\in A}\cos(ax)\leq -|A|^{1/10-o(1)}.$$ Another application makes significant progress on the problem of MaxCut in $H$-free graphs initiated by Erdős and Lovász in the 1970's. We show that every $m$-edge graph with no clique of size $m^{1/2-δ}$ has a cut of size at least $m/2+m^{1/2+\varepsilon}$ for some $\varepsilon=\varepsilon(δ)>0$.

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arXiv2025

Long induced paths in $K_{s, s}$-free graphs

More than 40 years ago, Galvin, Rival and Sands showed that every $K_{s, s}$-free graph containing an $n$-vertex path must contain an induced path of length $f(n)$, where $f(n)\to \infty$ as $n\to \infty$. Recently, it was shown by Duron, Esperet and Raymond that one can take $f(n)=(\log \log n)^{1/5-o(1)}$. In this note, we give a short self-contained proof that a $K_{s, s}$-free graphs with an $n$-vertex path contains an induced path of length at least $(\log \log n)^{1-o(1)}$. Combined with the recent remarkable example of Couëtoux, Defrain, and Raymond, which provides an upper bound of $O((\log \log n)^{1+o(1)})$, this essentially resolves this old problem.

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arXiv2025

Beyond the MaxCut problem in $H$-free graphs

In a recent breakthrough, Zhang proves that if $G$ is an $H$-free graph with $m$ edges, then $G$ has a cut of size at least $m/2+c_Hm^{0.5001}$, making a significant step towards a well known conjecture of Alon, Bollobás, Krivelevich and Sudakov. We show that the methods of Zhang can be further boosted, and prove the following strengthening. If $G$ is a graph with $m$ edges and no clique of size $m^{1/2-δ}$, then $G$ has a cut of size at least $m/2+m^{1/2+\varepsilon}$ for some $\varepsilon=\varepsilon(δ)>0$. In addition, we sharpen another result of Zhang by proving that if $G$ is an $n$-vertex $m$-edge graph with MaxCut of size at most $m/2+n^{1+\varepsilon}$ (or its smallest eigenvalue $λ_n$ satisfies $|λ_n|\leq n^{\varepsilon}$), then $G$ is $n^{-\varepsilon}$-close to the disjoint union of cliques for some absolute constant $\varepsilon>0$.

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arXiv2025

Factorization norms and an inverse theorem for MaxCut

We prove that Boolean matrices with bounded $γ_2$-norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We also present further structural results about Boolean matrices of bounded $γ_2$-norm and discuss applications in communication complexity, operator theory, spectral graph theory, and extremal combinatorics. As a key application, we establish an inverse theorem for MaxCut. A celebrated result of Edwards states that every graph $G$ with $m$ edges has a cut of size at least $\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}$, with equality achieved by complete graphs with an odd number of vertices. To contrast this, we prove that if the MaxCut of $G$ is at most $\frac{m}{2}+O(\sqrt{m})$, then $G$ must contain a clique of size $Ω(\sqrt{m})$.

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arXiv2025

Kővári-Sós-Turán theorem for hereditary families

The celebrated Kővári-Sós-Turán theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose the systematic study of this phenomenon, and state the conjecture that if $H$ is a bipartite graph, then an induced $H$-free and $K_{s,s}$-free graph cannot have much more edges than an $H$-free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by $k$ on one side, obtaining in all the cases similar bounds as in the non-induced setting. Our results also have applications to the Erdős-Hajnal conjecture, the problem of finding induced $C_4$-free subgraphs with large degree and bounding the average degree of $K_{s, s}$-free graphs which do not contain induced subdivisions of a fixed graph.

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arXiv2025

Factorization norms and Zarankiewicz problems

The $γ_2$-norm of Boolean matrices plays an important role in communication complexity and discrepancy theory. In this paper, we study combinatorial properties of this norm, and provide new applications, involving Zarankiewicz type problems. We show that if $M$ is an $m\times n$ Boolean matrix such that $γ_2(M)<γ$ and $M$ contains no $t\times t$ all-ones submatrix, then $M$ contains $O_{γ,t}(m+n)$ one entries. In other words, graphs of bounded $γ_2$-norm are degree bounded. This addresses a conjecture of Hambardzumyan, Hatami, and Hatami for locally sparse matrices. We prove that if $G$ is a $K_{t,t}$-free incidence graph of $n$ points and $n$ homothets of a polytope $P$ in $\mathbb{R}^d$, then the average degree of $G$ is $O_{d,P}(t(\log n)^{O(d)})$. This is sharp up the $O(.)$ notations. In particular, we prove a more general result on semilinear graphs, which greatly strengthens the work of Basit, Chernikov, Starchenko, Tao, and Tran.

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arXiv2024

Disjoint pairs in set systems and combinatorics of low rank matrices

We study and solve several problems in two closely related settings: set families in $2^{[n]}$ with many disjoint pairs of sets and low rank matrices with many zero entries. - More than 40 years ago, Daykin and Erdős asked for the maximum number of disjoint pairs of sets in a family $F\subseteq 2^{[n]}$ of size $2^{(1/2+δ)n}$ and conjectured it contains at most $o(|F|^2)$ such pairs. This was proven by Alon and Frankl in 1985. In this paper we completely resolve this problem, proving an optimal dependence of the number of disjoint pairs on the size of family $F$. We also prove the natural variant of the Daykin-Erdős conjecture in which disjoint pairs are replaced by pairs with intersection $λ\neq 0$. - Motivated by a conjecture of Lovett related to the famous log-rank conjecture, Singer and Sudan asked to show that for two families $A, B \subseteq 2^{[n]}$ with a positive constant fraction of set pairs $(a,b)\in A\times B$ being disjoint, there are $R\subset A$ and $S\subset B$ such that all set pairs $(r, s)\in R\times S$ are disjoint, and $|R|\geq 2^{-O(\sqrt{n})}|A|$ and $|S|\geq 2^{-O(\sqrt{n})}|B|$. We prove this conjecture in a strong quantitative form. - We prove the following generalizations of the best known bounds for the log-rank conjecture. If $M$ is an $n\times n$ non-negative integer matrix of rank $r$ in which the average of the entries is $\varepsilon\leq 1/2$, then $M$ contains an all-zero submatrix of size at least $2^{-O(\sqrt{\varepsilon r})}n$. Unlike the known bounds for the log-rank conjecture, this result is optimal. Moreover, using similar methods, we also prove that any $n\times n$ matrix of rank $r$ with entries from $\{0,\dots,t\}$ contains a constant submatrix of size at least $2^{-O(t\sqrt{r})}n$. Our proofs use probabilistic, entropy and discrepancy methods and explore connections to additive combinatorics and coding theory.

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arXiv2024

Large Cuts in Hypergraphs via Energy

A simple probabilistic argument shows that every $r$-uniform hypergraph with $m$ edges contains an $r$-partite subhypergraph with at least $\frac{r!}{r^r}m$ edges. The celebrated result of Edwards states that in the case of graphs, that is $r=2$, the resulting bound $m/2$ can be improved to $m/2+Ω(m^{1/2})$, and this is sharp. We prove that if $r\geq 3$, then there is an $r$-partite subhypergraph with at least $\frac{r!}{r^r} m+m^{3/5-o(1)}$ edges. Moreover, if the hypergraph is linear, this can be improved to $\frac{r!}{r^r} m+m^{3/4-o(1)},$ which is tight up to the $o(1)$ term. These improve results of Conlon, Fox, Kwan, and Sudakov. Our proof is based on a combination of probabilistic, combinatorial, and linear algebraic techniques, and semidefinite programming. A key part of our argument is relating the energy $\mathcal{E}(G)$ of a graph $G$ (i.e. the sum of absolute values of eigenvalues of the adjacency matrix) to its maximum cut. We prove that every $m$ edge multigraph $G$ has a cut of size at least $m/2+Ω(\frac{\mathcal{E}(G)}{\log m})$, which might be of independent interest.

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arXiv2024

Bisection Width, Discrepancy, and Eigenvalues of Hypergraphs

A celebrated result of Alon from 1993 states that any $d$-regular graph on $n$ vertices (where $d=O(n^{1/9})$) has a bisection with at most $\frac{dn}{2}(\frac{1}{2}-Ω(\frac{1}{\sqrt{d}}))$ edges, and this is optimal. Recently, this result was greatly extended by Räty, Sudakov, and Tomon. We build on the ideas of the latter, and use a semidefinite programming inspired approach to prove the following variant for hypergraphs: every $r$-uniform $d$-regular hypergraph on $n$ vertices (where $d\ll n^{1/2}$) has a bisection of size at most $$\frac{dn}{r}\left(1-\frac{1}{2^{r-1}}-\frac{c}{\sqrt{d}}\right),$$ for some $c=c(r)>0$. This bound is the best possible up to the precise value of $c$. Moreover, a bisection achieving this bound can be found by a polynomial-time randomized algorithm. The minimum bisection is closely related to discrepancy. We also prove sharp bounds on the discrepancy and so called positive discrepancy of hypergraphs, extending results of Bollobás and Scott. Furthermore, we discuss implications about Alon-Boppana type bounds. We show that if $H$ is an $r$-uniform $d$-regular hypergraph, then certain notions of second largest eigenvalue $λ_2$ associated with the adjacency tensor satisfy $λ_2\geq Ω_r(\sqrt{d})$, improving results of Li and Mohar.

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arXiv2024

The Quantitative Fractional Helly theorem

Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Bárány, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family $\mathcal{F}$ of $n$ convex sets in $\mathbb{R}^d$ such that at least $α\binom{n}{d+1}$ of the $(d+1)$-tuples of $\mathcal{F}$ have an intersection of volume at least 1, then one can select $Ω_{d,α}(n)$ members of $\mathcal{F}$ whose intersection has volume at least $Ω_d(1)$. Furthermore, with the help of this theorem, we establish a quantitative version of the $(p,q)$ theorem of Alon and Kleitman. Let $p\geq q\geq d+1$ and let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^d$ such that among any $p$ elements of $\mathcal{F}$, there are $q$ that have an intersection of volume at least $1$. Then, we prove that there exists a family $T$ of $O_{p,q}(1)$ ellipsoids of volume $Ω_d(1)$ such that every member of $\mathcal{F}$ contains at least one element of $T$. Finally, we present extensions about the diameter version of the Quantitative Helly theoerm.

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