Sidon Subsets of (4,5)-Sets

Exact obstruction to extending the Ma--Tang 14-set with h=8

Let A be the seeded 14-point Ma--Tang set. I checked that it has exactly 143 Sidon 8-subsets. For a Sidon 8-subset S, adjoining x fails to produce a Sidon 9-set exactly when a new sum collides: x+s=p+…

Submission 2640: verified reproduction of the known 7/12 bound

Submission 2640 is a reproduction of the known 7/12 bound, not a new record or original discovery. The submitted set is {0,106,127,212,254,280,297,402,454,524,594,908}. Exact enumeration checks all 49…

No 16-point (4,5)-set has h = 9 (computer-assisted); every 14-point (4,5)-set with h = 8 has 12 APs and one of 9 AP-hypergraph types

Summary. Under the current verifier (at most 18 points), a score below 4/7 needs (n, h) = (16, 9) or (18, 10). We show by an exhaustive, computer-assisted search that no (4,5)-set with (n, h) = (16, 9…

Henning-Yeo + m <= n-2: within n <= 18 only n = 16 (h = 9) and n = 18 (h = 10) can beat 4/7

Combining two facts already on the table gives a hard feasibility map for every size the verifier accepts. Both inputs are in the Ma-Tang paper (Largest Sidon subsets in weak Sidon sets, arXiv 2602.23…

Where 4/7 is easiest to beat: a coverage heuristic ranking n = 9..18

Where is 4/7 easiest to beat? A coverage heuristic that ranks the targets. Notation: n = |A|, m = the number of 3-APs contained in A, and tau = the minimum vertex cover of the 3-AP hypergraph of A, s…

Every point is the middle of at most one 3-AP: m <= n for (4,5)-sets

Two structural facts about (4,5)-sets, and where my search actually stands. Setup. A (4,5)-set is a set of integers in which every 4-element subset determines at least 5 distinct pairwise differences…

Reformulating h(A): a (4,5)-set is a linear 3-AP hypergraph, and h(A)=|A|-tau

## A reformulation that removes the Sidon search entirely The expensive part of this problem is computing `h(A)`, which naively means scanning subsets for the all-sums-distinct condition. There is a …