We prove the Erd\H{o}s--Chv\'atal simplex conjecture across its entire parameter range. If $2\le d<k$ and $(d+1)k\le dv$, then every $d$-simplex-free family of $k$-subsets of $[v]$ has at most $\binom{v-1}{k-1}$ members, with equality only for a full star.
In 1985, Erd{\H{o}}s, Frankl and F{\"u}redi conjectured that the threshold $T(r)$ for the existence of an $r$-cover-free family with more members than points satisfies $T(r)=(1+o(1))r^2$. We confirm this conjecture by proving $T(r)\ge r^2+r+2$ for every $r\ge1$.
Erd\H{o}s and S\'os conjectured that every $n$-vertex $3$-uniform hypergraph whose link graphs are all bipartite has at most $(1/4+o(1))\binom n3$ edges. We disprove this conjecture by constructing counterexamples with edge density at least $0.250000356>1/4$ for all sufficiently large $n$.
The Erd\H{o}s--Dushnik--Miller theorem states that for every aleph $\kappa$, \[ \kappa\to(\kappa,\omega); \] that is, every coloring $c:[\kappa]^2\to2$ has either a $0$-homogeneous set of cardinality $\kappa$ or a $1$-homogeneous set of cardinality $\omega$. In this article, we present a purely combinatorial proof of t...
The Erd\H os--S\'os conjecture states that every $n$-vertex graph with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We prove that for every $\gamma$ there is an $n_0$ such that for all $n\ge n_0$ and $k \ge \gamma n$ the conjecture holds. As a corollary of our result, we obtain a solution of a 51-year-old...
We present an exposition of a proof, discovered by GPT-6 Astra, of the Erd\H{o}s--S\'os Conjecture, which states that every graph with average degree greater than $t-2$ contains every tree on $t\geq 2$ vertices.
In the 1960s, Erd\H{o}s and Gallai conjectured that the edges of every $n$-vertex graph can be decomposed into $O(n)$ cycles and edges. We prove this conjecture. Equivalently, every Eulerian graph on $n$ vertices can be decomposed into $O(n)$ cycles, which confirms Haj\'os'conjecture up to a constant factor.
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